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+Can the double-slit experiment distinguish between quantum interpretations?
+Ali Ayatollah Rafsanjani,1, 2, ∗ MohammadJavad Kazemi,3, † Alireza Bahrampour,1, 3 and Mehdi Golshani2
+1Department of Physics, Sharif University of Technology, Tehran, Iran
+2School of Physics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
+3Research center for quantum engineering and photonics technology, Sharif University of Technology, Tehran, Iran
+Despite the astonishing successes of quantum mechanics, due to some fundamental problems
+such as the measurement problem and quantum arrival time problem, the predictions of the theory
+are in some cases not quite clear and unique. Especially, there are various predictions for the joint
+spatiotemporal distribution of particle detection events on a screen, which are derived from different
+formulations and interpretations of the quantum theory. Although the differences are typically small,
+our studies show that these predictions can be experimentally distinguished by an unconventional
+double-slit configuration, which is realizable using present-day single-atom interferometry.
+I.
+INTRODUCTION
+In textbook quantum theory, time is a parameter in the
+Schr¨odinger equation, not a self-adjoint operator, hence
+there is no unique and unambiguous way to compute the
+temporal probability distribution of events from the first
+principles (i.e.
+the Born rule) [1].
+Nonetheless, since
+clocks exist and time measurements are routinely per-
+formed in quantum experiments [2, 3], a complete quan-
+tum theory must be able to predict the temporal statis-
+tics of detection events. For example, in the famous dou-
+ble slit experiment, each particle is detected at a ran-
+dom time as same as at a random position on the de-
+tection screen [4–8].
+Therefore, one can ask: What is
+the position-time joint probability density P(x, t) on the
+screen? Although this question is very old [9–12], it is
+still open [13–18]. In fact, the ambiguity in the arrival
+time distribution even prevents a clear prediction of cu-
+mulative arrival position distribution,
+�
+P(x, t)dt, which
+is typically measured in a non-time-resolved double-slit
+experiment [19].
+Nonetheless, usual experiments are performed in the
+far-field (or scattering) regime, where a semiclassical
+analysis is often sufficient [13, 19]. In this analysis, it is
+assumed that particles move along classical trajectories,
+and the arrival time distribution is computed using the
+quantum momentum distribution [8, 20, 21]. However,
+because of the quantum backflow effect [22], even in free
+space, the quantum mechanical time evolution of position
+probability density is not consistent with the underlying
+uniform motion assumption, especially in near-field inter-
+ference phenomena [23]. In fact, due to recent progress in
+the ultra-fast detectors technology (e.g. see [24–27]), it
+will be soon possible to investigate the near-field regime,
+where the semiclassical approximation breaks down and
+a deeper analysis would be demanded [13, 28, 29].
+To remedy this problem, based on various interpre-
+tations and formulations of quantum theory, several at-
+tempts have been made to introduce a suitable arrival
+∗ [email protected]
+† [email protected]
+time distribution.
+On the one hand, according to the
+(generalized) standard canonical interpretation, the ar-
+rival distribution is considered as a generalized observ-
+able, which is described by a positive-operator-valued
+measure (POVM), satisfying some required symmetries
+[10, 11, 30, 31].
+On the other hand, in the realistic-
+trajectory-based formulations of quantum theory, such
+as the Bohmian mechanics [32], Nelson stochastic me-
+chanics [33], and many interacting worlds interpretation
+[34], the arrival time distribution could be obtained from
+particles trajectories [7, 18, 35, 36]. Moreover, in other
+approaches, the arrival time distribution is computed via
+phenomenological modeling of the detection process, such
+as the (generalized) path integral formalism in the pres-
+ence of an absorbing boundary [12, 37–39], Schr¨odinger
+equation with complex potential or absorbing boundary
+[40–44], and so on [45–47].
+In principle, the results of these approaches are dif-
+ferent. However, in most of the experimental situations,
+the differences are typically slight, and so far as we know,
+in the situation where differences are significant, none of
+the proposals have been backed up by experiments in a
+strict manner [8, 36]. An experiment that can probe these
+differences would undoubtedly enrich our understanding
+of the foundations of quantum mechanics. The purpose
+of the present paper is to make it evident, via numerical
+simulations, that the famous two-slit experiment could be
+utilized to investigate these differences if we simply use
+a horizontal screen instead of a vertical one: see Fig. 1.
+Using current laser cooling and magneto-optical trapping
+technologies, this type of experiment can be realized by
+Bose-Einstein condensates, as a controllable source of co-
+herent matter waves [48–50]. Moreover, our numerical
+study shows that the required space-time resolution in
+particle detection is achievable using fast single-atom de-
+tectors, such as the recent delay-line detectors described
+in [51, 52] or the detector used in [6, 53].
+The structure of this paper is as follows: In Section
+II, we study the main proposed intrinsic arrival distri-
+butions. Then, in section III we compare them in the
+double-slit setup with vertical and horizontal screens and
+in different detection schemes. In Section IV, we study
+the screen back-effect, and we summarize in section V.
+arXiv:2301.02641v1  [quant-ph]  6 Jan 2023
+
+2
+II.
+“INTRINSIC” ARRIVAL DISTRIBUTIONS
+In this section, we first review the semi-classical ap-
+proximation and then scrutinize two main proposed in-
+trinsic arrival time distributions [16, 36] and their asso-
+ciated screen observables. In these approaches, the effect
+of the detector’s presence on the wave function evolution,
+before particle detection, is not considered. We discuss
+this effect in section IV.
+A.
+Semiclassical approximation
+As mentioned, in the experiments in which the detec-
+tors are placed far away from the support of the initial
+wave function (i.e.
+far-field regime), the semiclassical
+arrival time distribution is routinely used to the descrip-
+tion of the particle time-of-flight [21, 54–57]. In this ap-
+proximation, it is assumed that particles move classically
+between the preparation and measurement. In this ap-
+proach, the arrival time randomness is understood as a
+result of the uncertainty of momentum, and so the arrival
+time distribution is obtained from momentum distribu-
+tion [13, 17, 36, 58].
+In the one-dimensional case, the
+classical arrival time is given by
+t = m(L − x0)/p0,
+(1)
+which is applicable for a freely moving particle of mass
+m that at the initial t = 0 had position x0 and momen-
+tum p0 arriving at a distant point L on a line. Hence,
+for a particle with the momentum wave fuction ˜ψ0(p),
+assuming ∆x0 ≪|L − ⟨x⟩0|, the semiclassical arrival time
+distribution reads [58]
+ΠSC(t|x=L) = mL
+t2 | ˜ψ0(mL/t)|2.
+(2)
+This analysis could be generalized in three-dimensional
+space. Then, the distribution of arrival time at a screen
+surface S is given by [36]
+ΠSC(t|x∈S) = m3
+t4
+�
+S
+| ˜ψ0(mx/t)|2 x · dS,
+(3)
+where the dS is the surface element directed outward.
+The other main distribution that should be demanded
+is the joint position-time probability distribution on the
+screen, which is also called ”screen observable” [11]. Us-
+ing the conditional probability definition, the joint prob-
+ability of finding the particle in dS and in a time in-
+terval [t, t+dt] could be written as P(x, t|x ∈ S)dSdt =
+[Π(t|x∈S)dt] × [P(x|x∈S, t)dS] . In this regard, one can
+use the fact that ψt(x) is the state of the system, con-
+ditioned on the time being t in the Schr¨odinger picture.
+This implies that |ψt(x)|2 refers to the position probabil-
+ity density conditioned at a specific time t [14, 15, 59].
+Therefore, in the semiclassical approximation, the joint
+spatiotemporal probability density reads as
+PSC(x, t|x∈S) = NSCΠSC(t|x∈S) |ψt(x)|2
+(4)
+in which NSC ≡1/
+�
+S dS |ψt(x)|2 is the normalization con-
+stant, and dS =n·dS, where n is the outward unit normal
+vector at x∈S.
+B.
+“Standard” approach
+The first attempts to investigate the arrival time prob-
+lem, based on the standard rules of quantum theory, were
+made at the beginning of the 1960s by Aharonov and
+Bohm [60], and also Paul [61]. This approach starts with
+a symmetric quantization of classical arrival time expres-
+sion (1), as follows [62]:
+ˆtAB = mL ˆp −1 − m
+2 (ˆp −1 ˆx + ˆx ˆp −1),
+(5)
+where ˆx and ˆp=−i ∂/∂x are the usual position and mo-
+mentum operators, respectively, and ˆtAB is called the
+Aharonov-Bohm time operator. This operator satisfies
+the canonical commutation relation with the free Hamil-
+tonian operator, [ˆtAB, ˆp2/2m] = iℏ, which has been used
+to establish the energy-time uncertainty relation [63, 64].
+However, although ˆtAB is Hermitian (or symmetric in
+mathematics literature), it is not a self-adjoint operator
+[65]—a fact that is in agreement with Pauli’s theorem
+[1]. The origin of this non-self-adjointness can be under-
+stood as a result of the singularity at p = 0 in the mo-
+mentum representation, ˆtAB → (iℏm/2)(p−2 − 2p−1∂p)
+[65]. Nevertheless, although the (generalized) eigenfunc-
+tions of ˆtAB are not orthogonal, they constitute an over-
+complete set and provide a POVM, which are used to
+define the arrival-time distribution as follows [63, 65]:
+ΠSTD(t|x=L)=
+1
+2πℏ
+�
+α=±
+�����
+� ∞
+−∞
+dp θ(αp)
+�
+|p|
+m
+˜ψt(p)e
+i
+ℏ Lp
+�����
+2
+,
+(6)
+where θ(·) is Heaviside’s step function and ˜ψt(p) is the
+wave function in the momentum representation which
+could be obtained from the initial wave function ˜ψ0(p), as
+˜ψt(p) = ˜ψ0(p) exp
+�
+− itp2/2mℏ
+�
+. The distribution ΠSTD
+and its generalization in the presence of interaction po-
+tential have been referred to as the ”standard arrival-
+time distribution” by some authors [16, 66–69]. In fact,
+Grot, Rovelli, and Tate treated the singularity of (5) by
+symmetric regularization and obtained equation (6) via
+the standard Born rule [64]. The generalizations of equa-
+tions (5) and (6) in the presence of interaction potential
+have been investigated in various works [16, 31, 70–75].
+Using these developments, it has been shown that the
+non-self-adjointness of the free arrival time operator can
+also be lifted by spatial confinement [71, 76], and the
+above arrival time distribution could be derived from the
+limit of the arrival time distribution in a confining box
+as the length of the box increases to infinity [72]. Fur-
+thermore, recently, the distribution (6) is derived from a
+space-time-symmetric extension of non-relativistic quan-
+tum mechanics [77].
+
+3
+The three-dimensional generalization of (6) is derived
+by Kijowski’s [10] via an axiomatic approach. The as-
+sumed axioms are implied by the principle of the prob-
+ability theory, the mathematical structure of standard
+quantum mechanics, and the Galilei invariance [78].
+Based on these axioms, Kijowski constructed the follow-
+ing arrival time distribution for a free particle that passes
+through a two-dimensional plane S as
+ΠSTD(t|x ∈ S)
+=
+1
+2πℏ
+�
+α=±
+�
+R2d2p∥
+×
+�����
+� ∞
+−∞
+dp⊥ θ(αp.n)
+�
+|p⊥|
+m
+˜ψt(p)e
+i
+ℏ x.p⊥
+�����
+2
+,
+(7)
+where p⊥ ≡(p . n)n and p∥ ≡ p − p⊥ are perpendicular
+and parallel components of p relative to S respectively,
+and n is the outward normal of plane S.
+In fact, he
+first proves the above expression for the wave functions
+whose supports lie in the positive (or negative) amounts
+of p⊥. Then he uniquely derives the following self-adjoint
+variant of the (three-dimensional version of) Aharonov-
+Bohm arrival time operator, by demanding that the time
+operator be self-adjoint and leads to (7) for these special
+cases via the Born rule [10, 78]:
+ˆtL = sgn(ˆp⊥)
+�
+mLˆp−1
+⊥ − m
+2 (ˆp−1
+⊥ ˆx⊥ + ˆx⊥ˆp−1
+⊥ )
+�
+,
+(8)
+where ˆx⊥ ≡ ˆx.n and L (≡ x.n) represent the distance
+between the detection surface and the origin [29].
+Fi-
+nally, for an arbitrary wave function, the equation (7)
+could be derived from this self-adjoint operator. More-
+over, considering this time operator, besides the com-
+ponents of the position operator in the detection plane,
+ˆx∥ ≡ ˆx − (ˆx.n)n, Kijowski obtains the following expres-
+sion as the joint position-time distribution on the detec-
+tion screen via the Born rule [78]:
+PSTD(x, t|x∈S) =
+�
+α=±
+|ψα
+S (x, t)|2,
+(9)
+in which ψ±
+S (x, t) is the wave function on the basics of
+joint eigenstates of the operators ˆtL and ˆx∥. Explicitly
+ψ±
+S (x, t) =
+1
+(2πℏ)3/2
+�
+d3p θ(±p.n)
+�
+|p⊥|
+m
+˜ψt(p)e
+i
+ℏ x.p.
+(10)
+Note that, the arrival time distribution (7) could be re-
+produced by taking the integral of (9) over the whole of
+the screen plane. The joint space-time probability distri-
+bution (9), and its generalization for the particles with
+arbitary spin, have been also derived by Werner in an-
+other axiomatic manner [11]. Moreover, it is easy to see
+that the results (7) and (9) can be obtained from a reg-
+ularized version of the (three-dimensional generalization
+of) Aharonov-Bohm time operator, which is the same as
+the procedure used by Grot, Rovelli and Tate in one-
+dimensional cases [64].
+C.
+Quantum flux and Bohmian approach
+Inspiring by classical intuition, another proper candi-
+date for screen observables is the perpendicular compo-
+nent of the quantum probability current to the screen
+surface, J(x, t).n, where
+J(x, t) = − ℏ
+m Im [ψ∗
+t (x)∇ψt(x)] ,
+(11)
+and n is the outward normal to the screen S. This pro-
+posal is applicable for a particle in a generic external
+potential and a generic screen surface, not necessarily
+an infinite plane. There are several attempts to derive
+this proposal in various approaches, such as Bohmian
+mechanics for the scattering case in [79], decoherent his-
+tories approach in [80] as an approximation, or in [81] as
+an exact formula using the concept of extended probabil-
+ities, and so on [45, 46, 82]. Howover, even if the wave
+function contains only momentum in the same direction
+as n, the J(x, t) · n could be negative due to the back-
+flow effect [22]. This property is incompatible with the
+standard notion of probability. Nevertheless, this prob-
+lem could be treated from the Bohmian point of view:
+Using Bohmian trajectories, it can be shown that the
+positive and negative values of J(x, t) · n correspond to
+the particles that reach the point x at S in the same di-
+rection of n or the opposite direction of it, respectively
+[83, 84]. In this regard, through the Bohmian mechanics
+in one-dimension, Leavens demonstrates that the time
+distribution of arrival to x=L from both sides could be
+obtained from the absolute form of probability flux as
+[35, 85]
+ΠQF(t|x=L) =
+|J(L, t)|
+�
+dt |J(L, t)|,
+(12)
+which is free from the aforementioned problem.
+The three-dimensional justification of J(x, t) · n as an
+operational formulation of the arrival time model has
+been made in [82]. Also, the generalization of (12) for
+arrival to the surface S is given by [7, 13, 16, 86]
+ΠQF(t|x∈S) =
+�
+S dS|J(x, t)·n|
+�
+dt
+�
+S dS|J(x, t).n|,
+(13)
+with dS =n·dS the magnitude of the surface element dS
+which is directed outward at x ∈ S. To illustrate (13)
+and to generalize it to the case of joint arrival distri-
+bution, we can use the Bohmian point of view. In this
+theory, each particle has a specific trajectory, depending
+on the initial position, and so the rate of passing par-
+ticles through an area element dS centered at x ∈ S, in
+the time interval between t and t + dt, is proportional to
+ρt(x)|v(x, t)·dS|dt, where v(x, t)=J(x, t)/|ψt(x)|2 is the
+Bohmian velocity of the particle. Hence, using quantum
+equilibrium condition [87, 88], ρt(x) = |ψt(x)|2, and ac-
+complishing normalization, the joint arrival distribution
+could be represented by the absolute value of the current
+density as
+
+4
+PQF(x, t|x∈S) =
+|J(x, t)·n|
+�
+dt
+�
+S dS|J(x, t)·n|.
+(14)
+Now, by integrating (14) over all x ∈ S, we arrive at the
+three-dimensional arrival time distribution (13) for the
+screen surface S.
+It should be noted that Eq.
+(14) is
+not necessarily followed for an ensemble of classical par-
+ticles because a positive or negative current at a space-
+time point, (x, t), can in general have contributions from
+all the particles arriving to x at t from any direction.
+Nonetheless, since the Bohmian velocity field is single-
+valued, the particle trajectories cannot intersect each
+other at any point of space-time and so only a single tra-
+jectory contributes to the current density J(x, t) at the
+particular space-time point (x, t).
+Moreover, this fact
+implies that when v(x, t) · n>0 we can say that the tra-
+jectory and consequently the particle has passed through
+the screen from the inside and vice versa for v(x, t)·n<0.
+Hence, one can define the joint probability distribution
+for the time of arrival to each side of S as
+P±
+QF(x, t|x∈S) =
+J±(x, t)·n
+�
+dt
+�
+S dS J±(x, t)·n,
+(15)
+where J±(x, t) = ± θ(±J·n) J(x, t). In addition, note
+that there may be some trajectories which cross S more
+than once—and we have multi-crossing trajectories (see
+the typical Bohmian trajectory in Fig. 1). The course
+of the above inference to Eq. (14) was in such a manner
+that multi-crossing trajectories could contribute several
+times (see Fig. 2 (a)).
+However, one could assume the
+detection surface as a barrier that does not allow the
+crossed particle to return inside (see Fig. 2 (c)). In this
+case, it is suggested to use the truncated current defined
+as
+˜J(x, t) :=
+�J(x, t)
+if (x, t) is a first exit through S
+0
+otherwise
+(16)
+where (x, t) is a first exit event through the boundary
+surface S, if the trajectory passing through x at time t
+leaves inside S at this time, for the first time since t = 0
+[13, 79, 89]. The limiting condition in (16), imposes that
+the joint probability distribution based on it should be
+computed numerically using trajectories:
+˜PQF(x, t|x∈S) =
+˜J(x, t)·n
+�
+dt
+�
+S dS ˜J(x, t)·n
+.
+(17)
+Of course, the detection screen is not always a barrier-
+like surface (see Fig. 2 (b)), and one could assume that
+there is a point-like detector that lets the multi-crossing
+trajectories to contribute to the distribution and we can
+use (14) in such cases.
+Horizontal screen
+Vertical screen
+Ly
+Lx
+x
+y
+s
+o
+FIG. 1.
+Schematic double-slit experiment setup.
+The cen-
+ter of two slits is considered as the coordinate origin, and
+the vertical and horizontal screens are placed at x = Lx and
+y = Ly, respectively. The dashed black line shows a typical
+Bohmian trajectory that arrives at the horizontal screen. A
+suitable single-particle detector, in addition to particle arrival
+position, can record the arrival time using a proper clock.
+III.
+“INTRINSIC” SCREEN OBSERVABLE IN
+TWO-SLIT EXPERIMENT
+In this section, we study the discussed proposals in the
+previous section for the double-slit experiment. We com-
+pare the results of these proposals in the cases of vertical
+and horizontal screens (see Fig. 1), and also in different
+detection schemes. The main motivation for the study
+of the horizontal screen is the non-classical particles’ mo-
+tions along the y-direction, in the Bohmian perspective;
+see a typical Bohmian trajectory in Fig. 1. This behav-
+ior is due to changing the sign of the probability cur-
+rent’s component in the y-direction. This behavior does
+not occur for x-component of J and consequently for the
+Bohmian motion of a particle along the x-direction.
+As shown in Fig. 1, the setup contains two identical
+slits at y = ±s, and screens are placed at x = Lx and
+y=Ly correspond to the vertical and horizontal screens,
+respectively. To avoid the mathematical complexity of
+Fresnel diffraction at the sharp-edge slits, it is supposed
+that the slits have soft edges that generate waves hav-
+ing identical Gaussian profiles in the y-direction. So, for
+each slit, we can take the wave function as an uncorre-
+lated two-dimensional Gaussian wave packet, which in
+each dimension has the form
+ψ(i)
+G (x, t) = (2πs2
+t)- 1
+4 exp
+�
+(x − x(i)
+0 − uxt)2
+4σ0st
+�
+× exp
+� i
+ℏmux(x − x(i)
+0 − uxt
+2 )
+�
+(i = 1, 2),
+(18)
+with m the particle’s mass, σ0 the initial dispersion, ux
+
+5
+S
+S
+S
+(a)
+(b)
+(c)
+First-arrival
+Second-arrival
+Third-arrival
+FIG. 2. Different schemes of particle detection on the screen
+surface S. In the Bohmian point of view, particles could have a
+recursive motion on surface S and cross it more than once (e.g.
+see the trajectory that plotted in Fig. 1). Assuming different
+detector types, one can prob variant possible observables on
+the screen. In panel (a) a conceivable particle trajectory is
+depicted, which crosses S three times. In this panel, a movable
+point-like detector is placed on S, which can survey the whole
+screen and detect particles that arrive only from one side,
+while in panel (b) a two-sided point detector is placed on
+S, which can move along it and detect particles that arrive
+from up and down. In addition, one can assume there is (c)
+an array of side-by-side detectors covering the entire screen
+surface S. The last configuration blocks the trajectory and
+does not allow the crossed particle to return. In this scheme,
+we only detect first-arrivals from one side.
+the wave packet’s velocity, x(i)
+0
+the initial position of wave
+packet or in other words the location of i-th slit, and
+st = σ0(1 + iℏt/(2mσ2
+0)). Therefore, when the particle
+passes through the slits, we have the total wave function
+as
+ψ(x, y, t) =
+1
+√
+2[ψ(1)
+G (x, t)ψ(1)
+G (y, t) + ψ(2)
+G (x, t)ψ(2)
+G (y, t)],
+(19)
+where superscripts (1) and (2) correspond to upper and
+lower slits, respectively. This form of Gaussian superpo-
+sition state is commonly used in the literature [7, 90–93]
+and is feasible to implement by quantum technologies be-
+cause such a state could be produced and controlled read-
+ily [94, 95], even without using slits [49]. In this paper,
+we have chosen the metastable helium atom, with mass
+m = 6.64 × 10−27 kg, as the interfering particle, and the
+parameters as s = 10 µm, σx = 0.04 µm, σy = 0.5 µm,
+ux = 3 m/s, and uy = 0 m/s. These values are feasible
+according to the performed experiments [96]. Moreover,
+the meta-stable helium atom could be detected with high
+efficiency because of its large internal energy [52, 97].
+A.
+Vertical screen
+The arrival time distribution for the vertical screen
+placed at different distances from the two-slit is shown
+in Fig. 3. As one can see this distribution is the same for
+all methods, and their average arrival time is close to the
+corresponding quantity in classical uniform motion. To
+calculate the mean time of arrival to the screen, we use
+the arrival time distribution of each method presented in
+sec II, i.e., Eq. (3), (7) and (13), and we have
+¯tS =
+� ∞
+0
+dt Π(t|x∈S) t,
+(20)
+as the mean arrival time at the surface S. Furthermore,
+we can compute the average arrival time to each point
+on the screen using the joint probability distribution as
+¯tx =
+� ∞
+0
+dt P(x, t|x∈S) t
+� ∞
+0
+dt P(x, t|x∈S) .
+(21)
+This observable is depicted in Fig. 4-b for a vertical screen
+placed at Lx = 300 mm. Apparently, the results of the
+standard and quantum flux methods are the same and
+similar to one that resulted in [7] by Nelson’s mechanics.
+Nevertheless, they are different from the semiclassical ap-
+proximation. However, when the interference pattern is
+calculated by either method, we see that their predicted
+cumulative position distributions do not differ much from
+the others (Fig. 4-a). This observable can be calculated
+by using the joint distribution as
+▲▲▲▲▲▲▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲▲▲▲▲▲
+▲
+▲
+▲
+▲
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+70
+80
+90
+100
+110
+120
+130
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+▲
+▲
+▲
+▲
+▲
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+70
+80
+90
+100
+110
+120
+130
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+▲
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+70
+80
+90
+100
+110
+120
+130
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+70
+80
+90
+100
+110
+120
+130
+Lx =330 mm
+Lx =300 mm
+Lx =270 mm
+Lx =240 mm
+Π(t | x = Lx)
+t (ms)
+0
+0.08
+0
+0.08
+0
+0.08
+0
+0.08
+Semiclassical
+Quantum flux
+Standard
+FIG. 3. Arrival time distributions of particles that arrive at
+the vertical screen of the double-slit experiment at different
+screen distances.
+
+6
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+-4
+-2
+0
+2
+4
+▲
+▲
+▲
+▲▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲▲
+▲
+▲
+▲
+-4
+-2
+0
+2
+4
+94
+96
+98
+100
+102
+104
+106
+(a)
+(b)
+Averaged arrival time (ms)
+P(y)
+y (mm)
+0
+0.5
+Semiclassical
+Standard
+Quantum flux
+FIG. 4. (a) The cumulative arrival position distribution, Eq.
+(22), for the vertical screen at Lx = 300 mm, and (b) the
+average arrival time at each point of the screen, Eq. (21).
+P(x|x∈S)=
+� ∞
+0
+dt P(x, t|x∈S)
+� ∞
+0
+dt
+�
+S dS P(x, t|x∈S).
+(22)
+As mentioned, it should be noted that, |ψt(x)|2 is just the
+conditional position probability density at the specific
+time t, not the position-time joint probability density
+and so the accumulated interference pattern, P(x|x∈S),
+is not given by
+�
+dt|ψt(x)|2 [98].
+B.
+Horizontal screen
+In this section, we are going to compare the mentioned
+proposals in the double-slit setup with a horizontal de-
+tection screen (see Fig. 1). In this regard, in Fig. 5, the
+arrival time distributions at the screen are plotted for
+some horizontal screens which are located at Ly =15, 20,
+25, and 30 µm. In this figure, solid-black, dashed-green,
+and dash-dotted-blue curves represent the distributions
+ΠST D, ΠQF and ΠSC respectively.
+Also, the vertical
+lines show the average time of arrival to the screen, ¯tS,
+associated with these arrival time distributions.
+From
+this figure, one can see that, although the averages al-
+most coincide, the distributions are distinct. Moreover,
+as expected, when the screen’s distance from the center
+of the two slits Ly decreases, the difference between dis-
+tributions increases. Most of these differences occur in
+the early times, which are associated with the particles
+that arrive at the S in the near field. Furthermore, we
+observe that the ΠSC behaves quite differently from ΠQF
+and ΠST D. The distributions ΠQF and ΠST D are more
+or less in agreement, however, for the screen that is lo-
+cated at Ly =15 µm, a significant difference between the
+standard and quantum flux distributions occurs around
+t≈0.2 ms.
+0.1
+0.5
+1
+5
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+0.1
+0.5
+1
+5
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.1
+0.5
+1
+5
+10
+0.0
+0.2
+0.4
+0.6
+0.8
+0.1
+0.5
+1
+5
+10
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+Π(t | y = Ly)
+t (ms)
+Ly = 30 µm
+Ly = 25 µm
+Ly = 20 µm
+Ly = 15 µm
+Semiclassical
+Standard
+Quantum flux
+FIG. 5.
+Arrival time distributions of particles that arrive
+on the horizontal screen at four different distances from the
+center of two slits. The vertical lines show the average arrival
+time.
+To have a more comprehensive insight, we can look at
+the joint spatiotemporal arrival distributions in Fig. 6.
+In this figure, joint distributions, PSC, PSTD and PQF are
+plotted in three panels, for the horizontal screen surface
+located at Ly = 15 µm.
+These density plots clearly
+visualize differences between the mentioned arrival dis-
+tribution proposals. In these plots, we can see separated
+fringes with different shapes, which this fact imply
+that the particles arrive at the screen in some detached
+space-time regions. In the insets, one can see that the
+shapes of these regions are different for each proposal.
+In the joint density of the semiclassical approximation
+
+7
+(Fig.6-a), fringes are well-separated, while the standard
+distribution (Fig. 6-b) exhibits more continuity in its
+fringes. In addition, in the pattern of the quantum flux
+proposal (Fig. 6-c) there are grooves between every two
+fringes which is due to changing the sign of J(x, t) · n in
+(14). In all panels of Fig.6, the duration of “temporal
+no-arrival windows” between every two typical fringes
+variate in the range between 0.01 and 0.2 ms which
+has a spatial extension of about 0.3 to 2 mm.
+These
+space-time scales are utterly amenable empirically by
+current technologies [53, 96], which could be used to test
+these results.
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1
+2
+3
+4
+5
+6
+0.0
+0.5
+1.0
+1.5
+2.0
+0.02
+0.04
+0.08
+0.16
+0.32
+0.64
+1.28
+3.20
+4.00
+0.00
+0.02
+0.04
+0.08
+0.16
+0.32
+0.64
+1.28
+3.20
+5.00
+0.00
+0.02
+0.04
+0.08
+0.16
+0.32
+0.64
+1.28
+3.00
+0.00
+t (ms)
+t (ms)
+t (ms)
+x (mm)
+Quantum flux
+Standard
+Semiclassical
+(c)
+(b)
+(a)
+FIG. 6. Density plots of joint arrival position-time distribu-
+tions for particles that arrive at the horizontal screen of the
+double-slit experiment. Panels (a), (b), and (c) represent PSC,
+PSTD and PQF, respectively. Insets: Magnified contour plots
+of the joint distributions.
+The average time of arrival to each point of the screen
+and cumulative position interference pattern could be cal-
+culated as in the vertical screen case by Eqs. (21) and
+(22). In Fig. 7(a)-(b), these two quantities are shown for
+the horizontal screen which is placed at y = 15 µm. In
+contrast to the vertical screen, the cumulative position
+distribution of the semiclassical approximation is entirely
+separate from the two other proposals. The cumulative
+position distribution resulting from standard and quan-
+tum flux approaches have obvious differences from each
+other, as well.
+As one can see in Fig. 7(b), the aver-
+age arrival times are the same for all three methods at
+first and begin to deviate from each other at x ≈ 5 mm;
+then again, these curves converge to each other at x≈25
+mm, approximately. The maximum deviation between
+the standard and quantum flux average arrival time oc-
+curs at x≈19 mm, which is quite in the far-field regime—
+the width of the initial wave function is ∼ O(10−3)mm
+which is smaller than 19 mm. Therefore one can suggest
+the average arrival time in the gray region of Fig. 7(b) as
+a practical target for comparing these approaches experi-
+mentally. To this end, we study arrival time distributions
+at some points of this region as local arrival distributions.
+The arrival time distribution conditioned at a specific
+point x on the screen can be obtained as follow
+Πx(t|x∈S) =
+P(x, t|x∈S)
+� ∞
+0
+dt P(x, t|x∈S).
+(23)
+Using the associated joint distribution of each proposal,
+we have plotted Fig. 7(c)-(f) that show Πx(t|x ∈ S) at
+the positions x=16.2, 17.4, 18.4, 19.2 mm, on the screen
+placed at Ly = 15 µm.
+The broken black curves in
+Fig. 7 (c)-(f), resulting from the quantum flux proposal,
+against the smooth curves of the other two methods could
+be understood as the result of the changing the signa-
+ture of the y-component of the probability current: Note
+that, quantum flux distribution is given by the absolute
+value of the probability current. The origin of distinc-
+tions between the local average arrival times is more per-
+ceptible from these local arrival distributions. In princi-
+ple, these distributions could be probed using fast and
+high-resolution single-atom detectors [53, 97]. In partic-
+ular, the delay-line detector that is recently developed
+by Keller et al. [51] seems suitable for our purpose: It
+has the capability to resolve single-atom detection events
+temporally with 220 ps and spatially with 177µm at rates
+of several 106 events per second.
+We estimate by a numerical investigation that these lo-
+cal arrival distributions could be well reconstructed from
+about 104 number of detection events. As an example,
+in Fig. 7, the histograms associated with the probability
+densities of the panel (f) are plotted in panel (g), using
+104 numerical random sampling. It is easy to estimate
+that the recording of 104 particle detection events can de-
+termine the local average arrival time with a statistical
+error of about 10−2ms, while the differences between local
+average arrival times of various proposals are almost big-
+ger than 10−1ms. Using cumulative position distribution,
+
+0.90
+0.85
+0.80
+0.75
+0.70
+0.65
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+3.00.90
+0.85
+0.80
+0.75
+0.70
+0.65
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+3.00.90
+0.85
+0.80
+0.75
+0.70
+0.65
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+3.08
+5
+6
+7
+8
+5
+6
+7
+8
+5
+6
+7
+8
+5
+6
+7
+8
+5
+6
+7
+8
+x=19.2 mm
+x=18.4 mm
+x=17.4 mm
+x=16.2 mm
+Πx(t|x ∈ S)
+t (ms)
+∆N
+(g)
+(f)
+(e)
+(d)
+(c)
+0
+1.5
+0
+1.5
+0
+1
+0
+1
+0
+600
+Semiclassical
+Standard
+Quantum flux
+0
+5
+10
+15
+20
+25
+0
+2
+4
+6
+8
+3
+4
+5
+6
+7
+8
+1.0
+1.5
+2.0
+2.5
+0
+5
+12
+14
+16
+18
+20
+22
+24
+26
+0.001
+0.003
+0.005
+Averaged arrival time (ms)
+x (mm)
+P(x)
+(b)
+(a)
+0
+0.4
+FIG. 7. The space-time arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm. Panel (a)
+represents the average time of arrival at each point of the screen, ¯tx. Panel (b) represents the cumulative position probability den-
+sity. The panels (c)-(f) show the local arrival time probability densities, Πx(t|x∈S), at the at the points x=16.2, 17.4, 18.4, 19.2
+mm on the screen, which are chosen from the gray region in panel (b). The vertical lines in these panels represent the average
+arrival times. Panel (g) is Histograms associated with probability densities of panel (f), which are generated by 104 numerical
+random sampling.
+Fig. 7(b), one can estimate that, if the total number of
+particles that arrived at the screen is about 108, we have
+about 104 particles around x = 19.2 mm, in the spacial
+interval (19.1, 19.3). Using recent progress in laser cool-
+ing and magneto-optical trapping [97], the preparation
+of a coherent ensemble of metastable helium atoms with
+this number of particles is quite achievable [51].
+One might be inclined to think that the difference be-
+tween the quantum flux and standard average arrival
+times is just due to changing the signature of J(x, t) · n,
+but in the following, we show that even without the con-
+tribution of the negative part of J(x, t)·n, these proposals
+are significantly distinguishable: see Fig. 8.
+C.
+Detection schemes
+As we mentioned in section II C, according to the
+Bohmian deterministic point of view, there are several
+possible schemes to detect arrived particles, especially
+for the horizontal screen surface which we have recursive
+motions on it (see Fig. 1 and 2). One can assume that
+the horizontal screen is swept with a point-like detector
+that surveys all arrived particles at the surface S, which
+we call spot-detection scheme. In this scheme, one option
+is to use a unilateral detector to detect arrived particles
+at the top or bottom of S. In this case, the positive and
+negative parts of the quantum probability current have
+respectively corresponded to particles that arrive at the
+top or bottom of S (as shown in Fig. 2 (a)), and we must
+use Eq. (15) to calculate the screen observables. Addi-
+tionally, we can choose a bilateral detector (or two uni-
+lateral detectors) that prob all particles that arrive from
+both sides of S, along the time with several repeats of
+the experiment (as shown in Fig. 2 (b)). In these circum-
+stances (i.e. spot-detection scheme), there is no barrier
+in front of the particles before they reach the point of
+detection and we can use Eq. (14) to obtain the screen
+observables as in the two previous subsections.
+As we have already shown in section II C, whether the
+particles arrive from the top or bottom of S, the abso-
+lute value of the quantum probability current yield the
+trajectories’ density and consequently give the joint dis-
+tribution of the total arrival at each point of S.
+This
+fact is the case for the standard method, as well, how-
+ever, there is a subtle difference between the two propos-
+als in the spot-detection scheme. When we talk about
+
+9
+0
+5
+10
+15
+20
+25
+5
+20
+0
+2
+4
+6
+8
+0
+5
+10
+15
+20
+25
+0
+2
+4
+6
+8
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0
+1
+2
+3
+4
+5
+6
+0.0
+0.5
+1.0
+1.5
+2.0
+x (mm)
+t (ms)
+First arrivals
+Second arrivals
+Third arrivals
+All arrivals
+First arrivals
+Quantum flux
+FIG. 8. The space-time Bohmian arrival statistics for the double-slit experiment with a horizontal screen placed at Ly =15 µm.
+The interior curves in the central figure are the averaged times of arrival obtained by different detection schemes: see Fig. 2.
+The Left and top plots are marginal arrival time distributions and marginal arrival position distributions, respectively. The
+scatter plot is generated using 2 × 106 Bohmian trajectories, and the black, blue, and green points of the scatter plot represent
+the first, second, and third arrivals of Bohmian particles to the screen, respectively. The inset is a zoom-in of the dashed
+rectangle.
+the spot-detection in the Bohmian approach, it would
+be considered the possibility of multi-crossing and the
+distribution includes all-arrivals at S. Although, in the
+standard method there is an interpretation for ψ+
+S (x, t)
+and ψ−
+S (x, t) in Eq. (10), which relates them to the par-
+ticles arrive at S in a direction which is the same or op-
+posite with the direction of outward normal of the screen
+n, respectively [10, 64], nevertheless, since there are no
+defined paths in this approach, it is obscure whether it
+counts only the first-arrivals to each side of the screen or
+includes recursive movements of particles.
+Alternatively, along with the spot-detection scheme, it
+could be assumed that there is a continuous flat barrier
+in front of the particle’s paths as the detection surface
+or screen surface that does not allow particles to cross
+this surface. Depending on the screen’s length and posi-
+tion, there are several possibilities for the detection pro-
+cess. In each case, a specific number of particle paths
+contribute to the distribution of arrival time.
+In the
+simplest case, the screen blocks all the trajectories that
+reach the horizontal surface S, and we only detect the
+first-arrivals. In such a setup, we can no longer use the
+quantum flux method to represent Bohmian trajectories’
+first encounter with the surface; hence, the screen ob-
+servables must be obtained by numerical analysis, due to
+the definition of truncated current as in Eq. (16) and its
+corresponding joint distribution, ˜PQF(x, t|x∈S), defined
+in Eq. (17). By computing the Bohmian trajectories,
+we can find positions and times of the first-arrivals to
+the screen, and consequently calculate the arrival time
+distribution which mathematically could be defined as
+˜ΠQF(t|x∈S) =
+�
+S
+˜PQF(x, t|x∈S)dS.
+(24)
+Also, other observable quantities such as the cumulative
+spatial distribution and averaged arrival time over the
+detection surface could be defined and calculated numer-
+ically in a similar way—by substituting ˜PQF(x, t|x ∈ S)
+in Eqs. (21) and (22). Furthermore, we can complete the
+computations to find the second and third encounters to
+the surface (regardless of the barrier).
+In Fig. 8, we show our numerical results of Bohmian
+trajectories simulation. The background scatter plot is
+the position and time of arrivals of 2 × 106 trajectories.
+In this plot, the second and third arrivals are shown in
+blue and green, respectively. Here, it is more clear why
+
+10
+5
+6
+7
+8
+5
+6
+7
+8
+5
+6
+7
+8
+5
+6
+7
+8
+x=19.2 mm
+x=18.4 mm
+x=17.4 mm
+x=16.2 mm
+Π(t| x, y)
+t (ms)
+0
+1.5
+0
+1.5
+0
+1.2
+0
+1.5
+First arrivals
+Quantum flux
+All arrivals
+FIG. 9. Arrival time distribution at the horizontal screen po-
+sitions x = 16.2, 17.4, 18.4, 19.2 mm, and Ly = 15 µm, which
+are in the gray region of Fig (8). The width of sampling in
+each point is about δx = 0.25 mm, and 108 Bohmian trajec-
+tories are simulated to obtain these distributions.
+we interpret the grooves of the quantum flux density plot
+(Fig. 6 (c)) as a result of the multi-crossing of Bohmian
+trajectories.
+The three middle graphs are the average
+time of the first and all-arrivals, which are simulation re-
+sults of 108 trajectories, and are compared by the quan-
+tum flux method. As expected, the average time of all-
+arrivals fits on the quantum flux curve. However, the av-
+erage time of first-arrivals deviates from all-arrivals in the
+area discussed in the previous section (between x = 16.2
+mm and x = 19.2 mm).
+To scrutinize the deviation zone of Fig. 8 (the gray re-
+gion), Fig. 9 is drawn to show the arrival time distribu-
+tions of screen positions x = 16.2, 17.4, 18.4, 19.2 mm.
+As one can see, at the first recursive points of quantum
+flux distribution, the first-arrival distributions raise down
+to zero. This implies that in the presence of a barrier-
+like screen, there would be a big temporal gap between
+arrived particles. These gaps could be investigated as a
+result of the non-intersection property of Bohmian tra-
+jectories that cause a unilateral motion of particles along
+the direction of the probability current field.
+IV.
+SCREEN BACK-EFFECT
+In principle, the presence of the detector could mod-
+ify the wave function evolution, before the particle detec-
+tion, which is called detector back-effect. To have a more
+thorough investigation of detection statistics, we should
+consider this effect. Howsoever, due to the measurement
+problem and the quantum Zeno effect [9], a complete in-
+vestigation of the detector effects is problematic at the
+fundamental level, and it is less obvious how to model
+an ideal detector. Nonetheless, some phenomenological
+non-equivalent models are proposed, such as the gener-
+alized Feynman path integral approach in the presence
+of absorbing boundary [12, 37–39], Schr¨odinger equation
+with a complex potential [44], Schr¨odinger equation with
+absorbing (or complex Robin) boundary condition [40–
+44], and so on. The results of these approaches are not
+the same, and a detailed study of the differences is an in-
+teresting topic. In this section, we provide a brief review
+of the absorbing boundary rule (ABR) and path-Integral
+with absorbing boundary (PAB) models, then we com-
+pare them in the double-slit setup with the horizontal
+screen.
+A.
+Absorbing Boundary Rule
+Among the above-mentioned phenomenological mod-
+els, the absorbing boundary condition approach has the
+most compatibility with Bohmian mechanics [42]. The
+application of absorbing boundary condition in arrival
+time problem was first proposed by Werner [40], and re-
+cently it is re-derived and generalized by Tumulka and
+others using various methods [41–44]. Especially, it is re-
+cently shown that in a suitable (non-obvious) limit, the
+imaginary potential approach yields the distribution of
+detection time and position in agreement with the ab-
+sorbing boundary rule [44]. According to this rule, the
+particle wave function ψ evolves according to the free
+Schr¨odinger equation, while the presence of a detection
+screen is modeled by imposing the following boundary
+conditions on the Detection screen, x ∈ S,
+n · ∇ψ = iκψ,
+(25)
+where κ>0 is a constant characterizing the type of detec-
+tor, in which ℏκ/m represents the momentum that the
+detector is most sensitive to. This boundary condition
+ensures that waves with wave number κ are completely
+absorbed while waves with other wave numbers are partly
+absorbed and partly reflected [41, 99]. In the absorbing
+boundary rule, the joint spatiotemporal distribution of
+the detection event is given by quantum flux. Consider-
+ing (25), this distribution reads
+PABR(t, x|x∈S) =
+|ψABC|2
+�
+dt
+�
+S dS|ψABC|2 ,
+(26)
+where ψABC represent the solution of the free Schr¨odinger
+equation satisfying the aforementioned absorbing bound-
+ary condition.
+This distribution can be understood in
+terms of Bohmian trajectories.
+The Bohmian particle
+equation of motion, ˙X = (ℏ/m)Im [∇ψABC/ψABC], to-
+gether with the boundary condition (25), imply that tra-
+jectories can cross the boundary S only outwards and so
+there are no multi-crossing trajectories. If it is assumed
+
+11
+that the detector clicks when and where the Bohmian
+particle reaches S, the probability distribution of detec-
+tion events is given by (26), because the initial distribu-
+tion of the Bohmian particle is |ψABC(x, 0)|2 [41].
+B.
+Path-Integral with Absorbing Boundary
+In several papers [12, 37–39], Marchuwka and Schuss
+develop an interesting method to calculate the detec-
+tion effect of absorbing surface using the Feynman path
+integral method.
+They postulate a separation princi-
+ple for the wave function in which we could consider
+the (bounded wave function) as a sum of two parts,
+ψ(x, t) = ψ1(x, t) + ψ2(x, t), such that ψ1(x, t) corre-
+sponds to the survival part of the wave which is orthogo-
+nal to ψ2(x, t) at a time t and evolve independently [38].
+So, we can obtain the probability of survival of the parti-
+cle, denoted S(t), which is the probability of the particle
+not being absorbed by the time t, as
+�
+D d3x|ψ1(x, t)|2,
+where the integral is over the domain D, outside the ab-
+sorbing region.
+By discretizing the path integral in a
+time interval [0, t] and eliminating the trajectories that,
+in each time interval [t′, t′+∆t′] for all t′ < t, are reached
+to the absorbing surface S, the survival and consequently
+absorbing probability would be obtained. Based on this
+analysis, we could define a unidirectional probability cur-
+rent into the surface as d
+dt[1−S(t)], which yields a normal
+component of the multidimensional probability current
+density at any point on S as
+J(x, t)·n= λℏ
+mπ |n·∇ψ(x, t)|2
+× exp
+�
+− λℏ
+mπ
+� t
+0
+dt′
+�
+S
+dS|n·∇ψ(x′, t′)|2
+�
+,
+(27)
+where dS = n · dS is the magnitude of the surface ele-
+ment dS, n is the unit outer normal to the absorbing
+surface S, and λ is a proportionality factor with the di-
+mension of length [37, 62]. Also, ψ(x, t) is the solution
+of Schr¨odinger equation bounded and normalized in the
+domain D. Moreover, the normal component J(x, t)·n is
+supposed to be the probability density for observing the
+particle at the point x on the screen at time t [12, 39].
+C.
+Screen back-effect in two-slit experiment
+In order to complete the investigations carried out
+in section III, we are going to study the screen back-
+effect in the double-slit experiment with a horizontal
+screen.
+In this regard, we compare the arrival distri-
+butions which are resulted from the absorbing bound-
+ary rule (ABR), path-Integral with absorbing boundary
+(PAB), and Bohmian truncated current (BTC).
+We continue with the same initial conditions as in sec-
+tion III, and choose κ = 1 µm−1 for ABR. This value of
+0.0
+0.5
+1.0
+1.5
+2.0
+0.0
+0.5
+1.0
+1.5
+2.0
+0
+1
+2
+3
+4
+5
+6
+0.0
+0.5
+1.0
+1.5
+2.0
+0.02
+0.04
+0.08
+0.16
+0.32
+0.64
+1.28
+2.56
+5.12
+0.00
+0.02
+0.04
+0.08
+0.16
+0.32
+0.64
+2.50
+6.25
+12.50
+0.00
+0.02
+0.04
+0.08
+0.16
+0.32
+0.64
+1.28
+3.20
+6.00
+0.00
+t (ms)
+t (ms)
+t (ms)
+x (mm)
+Bohmian truncated current
+Path-Integral with Absorbing Boundary
+Absorbing Boundary Rule
+(c)
+(b)
+(a)
+FIG. 10.
+Density plots of joint probability distributions of
+position and time (screen observable) for the horizontal screen
+placed at y = 15 µm in the double-slit experiment.
+These
+densities are calculated by the three methods which take the
+screen effects into account.
+κ leads to the maximum absorption probability—which
+is almost 0.4—for the chosen initial wave function. In
+addition, for a more meaningful comparison, we consider
+λ = 1 µm in the PAB method, which leads to the same
+absorption probability as ABR. The resulting joint ar-
+rival time-position distributions of the three methods are
+depicted in Fig. 10. As one can see, the distributions of
+the ABR and PAB methods—i.e., panels (a) and (b) in
+Fig. 10—have more compatibility with each other than
+the result of the BTC method. However, there are dif-
+ferences between them which are more obvious in the
+zoomed areas. The joint density of the ABR is more uni-
+
+0.90
+0.85
+0.80
+0.75
+0.70
+0.65
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+3.00.90
+0.85
+0.80
+0.75
+0.70
+0.65
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+3.00.90
+0.85
+0.80
+0.75
+0.70
+0.65
+1.8
+2.0
+2.2
+2.4
+2.6
+2.8
+3.012
+▲
+▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲
+▲
+▲ ▲ ▲ ▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+0
+2
+4
+6
+8
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲ ▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲ ▲ ▲
+▲ ▲ ▲ ▲
+▲
+▲
+▲ ▲ ▲ ▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+2
+4
+6
+8
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲▲▲▲▲▲▲▲▲
+▲
+▲
+▲
+▲▲▲▲▲▲▲▲▲▲
+▲
+▲
+▲
+▲
+▲
+▲▲▲▲
+▲
+▲
+▲
+▲
+▲▲▲▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲
+▲
+▲▲
+▲▲
+▲
+▲▲▲▲
+▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲▲▲▲▲▲▲▲▲▲
+0
+Averaged arrival time (ms)
+x (mm)
+P(x)
+Π(t | y)
+0.0
+0.8
+2.5
+0
+Absorbing Boundary Rule
+Path-Integral with Absorbing Boundary
+Bohmian truncated current
+FIG. 11. Averaged time of arrival at each point of the screen
+(central figure), cumulative interference pattern (upper fig-
+ure), and distribution of time of arrival to the horizontal
+screen of the double-slit experiment placed at y = 15 µm
+(right-hand figure).
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+5
+6
+7
+8
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+5
+6
+7
+8
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲
+▲ ▲ ▲
+▲
+▲
+▲
+▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲ ▲
+▲
+▲ ▲ ▲
+▲
+▲ ▲
+▲ ▲
+▲
+▲
+▲
+▲ ▲ ▲
+▲ ▲ ▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+5
+6
+7
+8
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+▲ ▲ ▲ ▲ ▲ ▲
+▲ ▲ ▲
+▲
+▲ ▲
+▲
+▲
+▲ ▲ ▲
+▲
+▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲
+5
+6
+7
+8
+x=19.2 mm
+x=18.4 mm
+x=17.4 mm
+x=16.2 mm
+Π(t| x, y)
+t (ms)
+0
+1.5
+0
+1.5
+0
+1.2
+0
+1.5
+ABR
+PAB
+BTC
+FIG. 12. Arrival time distribution at the horizontal screen po-
+sitions x = 16.2, 17.4, 18.4, 19.2 mm, and Ly = 15 µm, which
+are calculated for the three methods which take the screen
+effects into account.
+formly distributed than of the PAB method. The empty
+areas between the fringes of the panel (c) of Fig. 10 are
+due to the elimination of the recursive trajectories—or
+in other words, are due to the elimination of second and
+third arrivals in Fig. 8.
+For a more detailed comparison, in Fig. 11 the spa-
+tial and temporal marginal distributions are shown. In
+addition, the associated local average arrival times are
+compared in the central panel of this figure. The PAB
+method leads to significant discrepancies in marginal dis-
+tributions; The maximum difference is about 40% that
+occurs around x≈0.8 mm, which seems testable clearly.
+In contrast to the previous results on intrinsic distri-
+butions, in which the difference between average arrival
+times was significant, there is a good agreement in this
+observable for the ABR and PAB methods.
+However,
+there is a significant difference between the average ar-
+rival time in these two methods and BTC around x = 6
+mm.
+In Fig. 12, the local arrival time distributions at
+some points on the screen are plotted, which show simi-
+lar behavior.
+V.
+SUMMARY AND DISCUSSION
+When and where does the wave function collapse? How
+one can model a detector in quantum theory? These are
+the questions that we investigated in this work. We tried
+to show that there is no agreed answer for these ques-
+tions, even for the double-slit experiment that has in it
+the heart of quantum mechanics [100]. This is a practical
+encounter with the measurement problem [73]. In this
+regard, we numerically investigated and compared the
+main proposed answers to these questions for a double-
+slit setup with a horizontal detection screen. It is shown
+that these proposals lead to experimentally distinguish-
+able predictions, thanks to the current single-atom de-
+tection technology.
+In this work, we suggest the meta-stable helium atom
+as a proper coherent source of the matter wave, however,
+other sources may lead to some practical improvements.
+For example, using heavier condensate atoms can lead
+to more clear discrepancies. Moreover, it is worth not-
+ing that although the experiment with photons may have
+some practical advantages, there are more complications
+in its theoretical analysis. This is partially because of the
+relativistic localization-causality problem [101–104]. The
+theoretical investigation of a proposed experiment for
+photons would be an interesting extension of the present
+work, which has been left for future studies.
+ACKNOWLEDGMENTS
+We sincerely thank Mohammad Hossein Barati for
+carefully reviewing the manuscript, and Sheldon Gold-
+stein for his helpful comments.
+
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+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+1
+The state-of-the-art 3D anisotropic intracranial
+hemorrhage segmentation on non-contrast head CT:
+The INSTANCE challenge
+Xiangyu Li, Gongning Luo, Kuanquan Wang, Hongyu Wang, Shuo Li, Jun Liu, Xinjie Liang, Jie Jiang,
+Zhenghao Song, Chunyue Zheng, Haokai Chi, Mingwang Xu, Yingte He, Xinghua Ma, Jingwen Guo, Yifan Liu,
+Chuanpu Li, Zeli Chen, Md Mahfuzur Rahman Siddiquee, Andriy Myronenko, Antoine P. Sanner, Anirban
+Mukhopadhyay, Ahmed E. Othman, Xingyu Zhao, Weiping Liu, Jinhuang Zhang, Xiangyuan Ma, Qinghui Liu,
+Bradley J MacIntosh, Wei Liang, Moona Mazher, Abdul Qayyum, Valeriia Abramova, Xavier Llad´o
+Abstract—Automatic intracranial hemorrhage segmentation in
+3D non-contrast head CT (NCCT) scans is significant in clinical
+practice. Existing hemorrhage segmentation methods usually
+ignores the anisotropic nature of the NCCT, and are evaluated
+on different in-house datasets with distinct metrics, making it
+highly challenging to improve segmentation performance and
+perform objective comparisons among different methods. The
+2022 intracranial hemorrhage segmentation on non-contrast head
+CT (INSTANCE 2022) was a grand challenge held in conjunc-
+tion with the 2022 International Conference on Medical Image
+Computing and Computer Assisted Intervention (MICCAI). It is
+intended to resolve the above-mentioned problems and promote
+the development of both intracranial hemorrhage segmentation
+and anisotropic data processing. The INSTANCE released a
+training set of 100 cases with ground-truth and a validation set
+with 30 cases without ground-truth labels that were available to
+the participants. A held-out testing set with 70 cases is utilized
+for the final evaluation and ranking. The methods from different
+participants are ranked based on four metrics, including Dice
+Similarity Coefficient (DSC), Hausdorff Distance (HD), Relative
+Volume Difference (RVD) and Normalized Surface Dice (NSD).
+A total of 13 teams submitted distinct solutions to resolve
+the challenges, making several baseline models, pre-processing
+strategies and anisotropic data processing techniques available to
+future researchers. The winner method achieved an average DSC
+of 0.6925, demonstrating a significant growth over our proposed
+baseline method. To the best of our knowledge, the proposed
+INSTANCE challenge releases the first intracranial hemorrhage
+segmentation benchmark, and is also the first challenge that
+intended to resolve the anisotropic problem in 3D medical image
+segmentation, which provides new alternatives in these research
+fields.
+Index Terms—Intracranial hemorrhage Segmentation Chal-
+lenge Anisotropic data
+I. INTRODUCTION
+I
+NTRACRANIAL hemorrhage (ICH) is a severe brain dis-
+ease and a main cause of stroke [1], [2]. It has a high
+mortality rate of 40% within one month [3], [4]. Furthermore,
+ICH even causes significant disability in survivor patients,
+with only 20% of patients expected to be capable of living
+independently in half year [5]. Therefore, early and accurate
+diagnosis of the ICH is important for saving patients’ lives
+and improve their prognosis in clinical practice [1], [6].
+Non-contract head computerized tomography (NCCT) is the
+primary imaging modality to diagnosing ICH for its widely
+availability in most emergency rooms and high sensitivity for
+detecting ICH. Moreover, NCCT enables accurate monitoring
+of hemorrhage progression, and effectively quantify hematoma
+volumes in ICH [1], [4], [7], making it a gold standard
+examination for the diagnosis of ICH.
+Hematoma volume estimation is significant for the prog-
+nosis and treatment decisions for ICH patients. In recent
+clinical trials, the hematoma volume has been utilized as
+a reliable indicator to determine the optimal candidates for
+intervention [8]–[10]. Thus, volume quantification of ICH has
+become an essential procedure for outcome predictions and
+ICH therapy. The hematoma volume can be estimated by
+semiautomated methods with the aid of radiologists, which
+is time-consuming [11] and suffers from inter-rater variability
+[12]. The ABC/2 method [13] is an effective technique to
+estimate hematoma volume in clinical practice since it is
+simple to implement. However, the estimation accuracy of the
+ABC/2 method dramatically decreases with irregular or large
+scale hemorrhages [8], [14]. The ICH segmentation methods,
+enabling accurate and rapid hematoma volume quantification,
+have become the leading criterion in ICH diagnosis.
+However, there exists plenty of challenges to segment ICH
+for automatic methods. For example, the hemorrhage struc-
+tures vary considerably across patients in terms of shape, size,
+and localization, preventing the use of valuable location and
+shape priors that are significant elements in the segmentation
+of many other anatomical structures. The blurred boundaries
+for the ICH region further improve the difficulty of the
+segmentation task [15].
+Because of the clinical significance and the intrinsic chal-
+lenges, the task of automatic intracranial hemorrhage segmen-
+tation has attracted extensive attention in the past few years.
+Recently, deep learning–based ICH segmentation models that
+segment ICH regions and quantify hematoma volume have
+been performed to effectively diagnose ICH and have achieved
+competitive results [6], [16]–[20]. However, all those above-
+mentioned ICH segmentation methods ignore the anisotropic
+nature of the NCCT volume by simply performing 2D or 3D
+convolutional networks, and they were evaluated on different
+in-house hemorrhage segmentation datasets with distinct met-
+rics, making it highly challenging to improve segmentation
+performance and perform objective comparisons among these
+arXiv:2301.03281v1  [eess.IV]  9 Jan 2023
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+2
+methods. Consequently, it remains hard to determine which
+kinds of segmentation techniques may be valuable to follow
+in clinical practice and research; what exactly the performance
+is of the state-of-the-art methods.
+To resolve the above-mentioned challenges on fair com-
+parisons of different methods, we organized the INtracranial
+hemorrhage SegmenTAtioN ChallengE on non-contrast head
+CT (INSTANCE) in conjunction with the 2022 international
+conference on Medical Image Computing and Computer As-
+sisted Interventions (MICCAI) in Singapore. To this end,
+we collected and released an ICH segmentation dataset of
+200 3D volumes with refined labeling from several experi-
+enced radiologists, and encouraged the participants to develop
+novel algorithms to effectively segment hematoma region with
+anisotropic NCCT volumes. Moreover, we evaluate different
+benchmark ICH segmentation methods with the same metrics,
+including Dice Similarity Coefficient (DSC), Hausdorff dis-
+tance (HD), relative volume difference (RVD) and normalized
+surface dice (NSD). Each of these benchmark methods was
+implemented by different challenge participants on a subset of
+the ICH dataset, and tested on a isolated testing dataset against
+the manually delineated groundtruth labels. To the best of our
+knowledge, INSTANCE is the first public intracranial hemor-
+rhage segmentation challenge, and also the first challenge that
+intended to deal with the anisotropic problem in 3D biomedical
+image segmentation. It is served as a solid benchmark for ICH
+segmentation tasks, and would also promote the development
+of intracranial hemorrhage segmentation and anisotropic data
+processing.
+II. PRIOR WORKS
+A. Related intracranial hemorrhage segmentation methods
+A large numbers of methods have been proposed to automat-
+ically segment intracranial hemorrhage in CT scans. Among
+them, deep learning techniques are widely adopted for its
+excellent performance in medical image segmentation tasks
+[15], [21]. Ironside et al. utilized U-Net [22] to automati-
+cally segment ICH and estimate the hematoma volume. They
+achieved comparable accuracy and greater efficiency compared
+to manual and semi-automated segmentation techniques [8].
+To address the issue of insufficient annotation data for ICH
+segmentation tasks, Kuo et al. proposed a patch-based FCN
+network and segmented ICH in an active learning manner [23].
+Chang et al. proposed an ROI-based framework that is opti-
+mized specifically for ICH detection and segmentation tasks by
+projecting 3D features to 2D networks in the feature pyramid
+network [18]. Kwon et al. proposed a Siamese U-Net method
+to segment ICH by leveraging the dissimilarity between
+learned features of healthy templates and input images [20].
+Kyung et al. proposed a supervised multi-task aiding represen-
+tation transfer learning network for ICH, which was divided
+into upstream and downstream. In the upstream, effective
+representation learning was performed by multi-task learning
+(classification, segmentation, reconstruction) and differences
+in the specific head of the consistency loss mitigation target are
+added. For downstream, feature extractor trained upstream is
+combined with 3D operator (classifier or divider) to implement
+specific tasks [16]. Wu et al. proposed a combination of an
+attention-based convolutional neural network and a variational
+Gaussian process for multiple instance learning method for
+predicting intracranial hemorrhage slices [24]. Toikkanen et
+al. proposed a residual segmentation method based on gener-
+ative adversarial network, which generates the image without
+bleeding in the original section through the model, and then
+calculates the difference between the generated image and the
+original image, so as to obtain the segmented image [17].
+Abramova et al. introduced the squeeze-excitation block into
+3D U-Net to solve the problem of segment hemorrhagic stroke
+lesions. Moreover, a restrictive patch sampling is proposed to
+alleviate the class imbalance problem and also to deal with
+the issue of intra-ventricular hemorrhage [25]. Kuang et al.
+designed new self-attention blocks and contextual attention
+blocks that take full advantage of both in-chip and inter-
+chip information. In addition, multilevel training strategies are
+proposed to reduce the influence of inter-class imbalance [26].
+Wang et al. propose a Masked Multi-Task Network method
+to detect brain CT volumes with intracranial hemorrhage and
+distinguish hemorrhage type by leveraging different types of
+intracranial hemorrhage at different locations [27]. Guo et al.
+propose a full convolutional neural network for simultaneous
+classification and segmentation of ICH, and the ConvLSTM
+module was used to address this issue of the loss of spatial
+information [28]. Kadam et al. propose architectures combined
+Xception and LSTM/GRU for classification of Intracranial
+Hemorrhage. It is also found through experiments that Xcep-
+tion GRU model has better performance on most of the metrics
+as compared to the Xception and Xception LSTM models [29].
+Despite the excelent results reported in the above pa-
+pers, it is still challenging to identify the best performing
+method among them because of the varied testing datasets
+and evaluation metrics. The proposed INSTANCE challenge
+provides a standardized procedure to systematically evaluate
+and compare different SOTA methods on the same testing
+dataset and consistent evaluation metrics, enabling objective
+and fair comparison among different techniques.
+B. Medical Image Segmentation Challenges
+Recently years have witnessed the growing popularity for
+biomedical image analysis challenges, especially for medical
+image segmentation challenges. To name a few, there were 25,
+20, and 40 accepted challenges at the International Conference
+on Medical Image Computing and Computer-Assisted Inter-
+vention (MICCAI) 2020, 2021, and 2022, respectively. From
+2020 to 2022, the number of challenges nearly doubled, and
+the segmentation-related challenges occupied 38% of all the
+challenges1. Similarly, in the largest biomedical image chal-
+lenge platform ’Grand Challenge2’, 149 out of 315 (47.3%)
+challenges are designed for segmentation tasks. There are lots
+of successful challenges in medical image segmentation, for
+example, the Brain Tumor Segmentation (BraTS) challenge
+[30] provide a solid benchmark for multimodal brain tumor
+1https://www.biomedical-challenges.org/
+2https://grand-challenge.org/
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+3
+segmentation task, numerous methods on brain tumor seg-
+mentation and multi-modal learning have been validated on
+this benchmark, significantly improving the development of
+those research fields. The Head and neck tumor segmentation
+challenge (Hecktor) [31] organized a novel challenge for head
+and neck tumor segmentation on PET/CT modalities, which
+claimed to be the pioneer work on this field. The abdomen ct
+organ segmentation [32] first consider the inference time, and
+GPU memory consumption as extra evaluation metrics instead
+of simply focusing on the segmentation accuracy, providing a
+novel benchmark with more comprehensive evaluation metrics.
+The Kidney Tumor Segmentation (KiTS) ( [33] ) challenge
+allow participants to compare their methods on kidney and
+kidney tumor segmentation tasks.
+Those above challenges have made great progress in pro-
+moting the development of specific medical field. However, to
+the best of our knowledge, there are no challenges intended
+to resolve the ICH segmentation with anisotropic 3D volumes.
+Hence, the INSTANCE is the first released grand challenge for
+the ICH segmentation and also the first challenge that intended
+to deal with the anisotropic problem in 3D medical image
+segmentation. We believe that the ICH data and algorithms
+provided in this benchmark would be helpful to promote
+the development of both ICH diagnosis and anisotropic data
+processing.
+III. THE ORGANIZATION OF THE INSTANCE CHALLENGE
+The proposed INSTANCE challenge was organized in 2022
+and was in conjunction with the 25rd MICCAI conference
+as a satellite event. It was deployed on the Grand Challenge
+platform. The official webpage of the INSTANCE challenge
+is https://instance.grand-challenge.org/. Meanwhile, we also
+construct a Github repository
+3 which provides plenty of
+resources related to the challenge, for example, the agree-
+ment files for accessing the dataset, the docker rules and
+submission examples, and also the baseline models. For the
+challenge schedule, the registration is open to the public on
+March 28, 2022. The training and validation dataset were
+released on April 6 and July 15, respectively. The dead-
+line of the open validation phase and the testing phase is
+on August 7 and August 14, respectively. In the validation
+phase, the participants uploaded their segmentation results to
+the Grand challenge website, and the platform automatically
+calculated the evaluation metrics by comparing them with
+the ground-truth labels we provided, and then displayed the
+calculated metrics on the validation leaderboard 4 In the testing
+phase, the participants are required to submit one successful
+docker image that contains their algorithms, and we ran the
+docker images from different participants on the closed testing
+dataset. The dataset of the INSTANCE challenge are currently
+available to the public on Grand Challenge platform after
+signing an agreement file and the post-challenge leaderboard
+submission is open for researches in this community. The
+following sections summarizes the detailed implementation of
+the INSTANCE challenge.
+3https://github.com/PerceptionComputingLab/INSTANCE2022
+4https://instance.grand-challenge.org/evaluation/challenge/leaderboard/
+A. Dataset
+We obtained the approval from Peking university, shougang
+hospital to perform a retrospective analysis of the patients that
+were diagnosed as intracranial hemorrhage between 2017 and
+2019. We then collected 200 non-contrast head CT volumes
+of those patients to construct challenge dataset. For these
+200 cases, they were diagnosed as different kinds of ICHs,
+including intraparenchymal hemorrhage (IPH), intraventricular
+hemorrhage (IPH), subarachnoid hemorrhage (SAH), subdural
+hemorrhage (SDH), and epidural hemorrhage (EDH), an exam-
+ple for each kind of ICH is illustrated in Fig. 1. We then split
+the 200 cases into training, validation and testing, with 100, 30,
+and 70 cases respectively. The CT scans and the labels of the
+training set are available to the participant for model training,
+while only the CT scans are provided for them to tune their
+algorithms on the Grand Challenge platform. Finally, in the
+testing phase, we provide a closed test set for fair comparison
+between different methods.
+For each of the subject in INSTANCE dataset, we first
+converted the traditional Digital Imaging and Communications
+in Medicine (DICOM) files to the Neuroimaging Informatics
+Technology Initiative (NIfTI) format. In this way, each subject
+only has one single NIfTI file instead of a bunch of DICOM
+files, making it easier to process in a image segmentation
+program. The volume sizes ranges from 512 × 512 × 20 to
+512 × 512 × 70, and the pixel spacing of a CT volume is
+0.42mm × 0.42mm × 5mm, hence the volume is anisotropic
+with inter-slice resolution much smaller than the within-slice
+resolution. The window width and the window center is 90HU
+and 40HU, respectively. We kept the original Hu value in
+the NIfTI volume since the participants can conduct different
+windowing strategies.
+For the data annotation, we gathered several experienced
+radiologists and some postgraduate students majored in med-
+ical imaging to perform hemorrhage region annotation in
+the NCCT scans. To improve the efficiency of the annota-
+tion process, we adopted a coarse-to-fine annotation strategy.
+Specifically, the ICH lesions were first manually delineated in
+the NCCT volumes with a popular annotation software in med-
+ical imaging, Seg3D5 [34]. Then the experienced radiologists
+checked the coarse annotations and manually refined them.
+Finally, all the radiologists double-check the annotations from
+other annotators, and discuss to achieve the final annotations
+with majority voting strategy.
+B. Evaluation Measures and Ranking Method
+The INSTANCE challenge adopted four accuracy-related
+evaluation metrics: Dice Similarity Coefficient (DSC), Haus-
+dorff Distance (HD), Relative Volume Difference (RVD) and
+Surface Dice (NSD) [35]. We utilized DSC and HD since
+they are widely used in different medical image segmentation
+challenges. They are complementary metrics for evaluating
+segmentation performance. DSC was utilized to measure the
+region overlapping error between ground truth and segmen-
+tation results, while HD is used to evaluate the coincidence
+5https://www.sci.utah.edu/cibc-software/seg3d.html
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+4
+Fig. 1: Different kinds of intracranial hemorrhages, including intraparenchymal hemorrhage (IPH), intraventricular hemorrhage
+(IPH), subarachnoid hemorrhage (SAH), subdural hemorrhage (SDH), and epidural hemorrhage (EDH). The varied shapes and
+positions for different kinds of hemorrhages promote the difficulties of the segmentation task.
+TABLE I: The Correspondence between the Team names and
+the aliases.
+Team
+Alias
+vegetable
+T1
+nvauto
+T2
+mec-lab
+T3
+ibot
+T4
+stubmers
+T5
+crainet
+T6
+superembrace
+T7
+scan
+T8
+dolphins
+T9
+nic-vicorob
+T10
+2i mtl
+T11
+avich
+T12
+visal
+T13
+between segmented surface and target surface. We used the
+RVD since the purpose for the ICH segmentation is to quantify
+the hematoma volume, making the volume differences be-
+tween the predictions and the labels significant for the results
+analysis. Moreover, we further added the NSD metric as a
+complement evaluation for the HD metric because the HD
+would become infinite when the prediction is a normal head
+CT scan without hemorrhages. The NSD also measures the
+discrepancy between the target and predicted boundaries.
+We intended to rank different algorithms based on the
+above-mentioned four metrics. Motivated from the former
+challenges [31], [36], we utilized a “aggregate-then-rank”
+scheme for ranking, including the following three steps: (1)
+Calculate the average DSC, HD, RVD and NSD metrics for
+all cases in the testing dataset. (2) Rank all the participant
+teams on these four metrics, hence each team would get four
+ranks. (3)Based on the rankings generated from (2), we then
+averaged these rankings and achieved the final ranking for each
+team. Moreover, for some extreme cases, e.g., the HD metric
+is infinite because the algorithm mistakenly treated some hard
+ICH cased as normal head scans. In this case, we treat all
+‘inf’ teams the same rank on HD which are inferior to others.
+Because we believe effectively diagnosis hard samples is also
+important in our challenge.
+IV. RESULTS
+A. Participation and submissions
+The INSTANCE 2022 received over 500 applications on
+grand-challenge platform and 70 teams were approved to be
+able to access the challenge dataset. The reason why we
+refused the other applications was that they didn’t submit the
+signed agreement files that we provided in the participation
+rules. In the validation phase, 30 teams uploaded their results
+with over 350 valid submissions on the grand challenge
+website. The final validation leaderboard is available on Grand
+challenge website. In the testing phase, 13 teams successfully
+submitted the Docker containers and the short papers.
+B. Algorithm summary
+We adopted the SLEX-NET [6] as the baseline model in
+the proposed INSTANCE challenge. It is noted that the dataset
+utilized in the SLEX-NET is different from INSTANCE 2022.
+Therefore, we re-trained the algorithm of baseline model
+on the INSTANCE 2022 dataset, with other training details
+consistent with the settings in the original paper.
+For the participants’ models, we find out that all the partic-
+ipants chose U-Net-related architectures, including attention
+U-Net [37], U-Net [22], 3D U-Net [38], nnU-Net [39], etc.
+Among them, nnUNet is still the most popular model, 7 out
+of 13 teams adopted it as their backbone network. Moreover,
+we also summarized other key factors in the methods by those
+participants, including data augmentation, loss functions, pre-
+processing, post-processing, and etc. The detailed summaries
+are illustrated in Table II. It shows that all teams used data
+augmentation, and 10 out of 13 teams conducted ensemble
+learning to improve their performance. In addition, four teams
+utilized the 2D implementation, seven teams adopted the
+3D implementation, and two teams combined 2D/3D imple-
+mentations. For the pre-processing and post-processing, all
+teams conducted different kinds of pre-processings, including
+normalization, windowing, skull-stripping, and etc, while only
+one team applied post-processing. To improve the learning
+of deep models, each team utilized different losses, such as
+Dice loss, cross-entropy loss, focal loss, and etc. Detailed
+descriptions of their methods can be found in the Appendix A.
+More importantly, we also released their submitted papers on
+
+IPH
+IVH
+SAH
+SDH
+EDHJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+5
+TABLE II: Summary of the algorithms in terms of key factors in the methods by those participants: backbone network,
+2D/3D, stages, pre-processing, data augmentation, loss functions, ensembles, post-processing. Abbreviation: Normalization
+(N), Windowing(W), Skull stripping(SS), Cropping (CP), Cross-Entropy(CE), Tversky (TV), Contour loss (CT)
+Team
+Backbone
+2D/3D
+Preprocess
+Stage
+Augmentation
+Loss
+Ensemble
+Postprocess
+Patch-based
+T1
+nnU-Net
+2D/3D
+N
+1
+✓
+Dice+WCE
+✓
+�
+✓
+T2
+ResUNet
+2D
+N
+2
+✓
+Dice+CE
+✓
+�
+�
+T3
+nnU-Net
+3D
+SS
+1
+✓
+Dice+CE+CT
+✓
+�
+�
+T4
+nnU-Net
+3D
+N+W+CP
+1
+✓
+Dice+CE
+�
+�
+✓
+T5
+nnU-Net
+3D
+N+SS
+1
+✓
+Dice+CE
+✓
+�
+✓
+T6
+nnU-Net
+3D
+N+W
+2
+✓
+Dice+Focal
+✓
+�
+✓
+T7
+nnU-Net
+3D
+N+W
+1
+✓
+Dice+CE
+✓
+�
+✓
+T8
+U-Net
+3D
+W
+1
+✓
+CE
+✓
+�
+✓
+T9
+nnU-Net
+2D/3D
+N
+2
+✓
+CE
+✓
+�
+✓
+T10
+U-Net
+3D
+N+SS
+1
+✓
+Dice+CE
+✓
+✓
+✓
+T11
+Attention U-Net
+2D
+N
+2
+✓
+Dice+CE+TV
+✓
+�
+�
+T12
+U-Net
+2D
+N
+1
+✓
+Dice
+�
+�
+�
+T13
+U-Net3+
+2D
+SS
+1
+✓
+Dice+CE
+�
+�
+�
+TABLE III: Summary of the INSTANCE 2022 validation phase. The average DSC, RVD, NSD and HD are reported for the
+baseline models and the submitted algorithms from each participant. The unit of HD is [mm]. Bold values represent the best
+scores for each metric.
+Team
+DSC(%)↑
+NSD(%)↑
+RVD↓
+HD↓
+T1
+79.12±23.00
+50.26±19.91
+0.21±0.20
+29.02±26.34
+T2
+78.21±18.45
+55.28±12.67
+0.20±0.18
+32.30±30.04
+T3
+71.60±30.10
+50.60±21.30
+0.29±0.30
+inf
+T4
+73.55±26.74
+51.57±18.10
+0.24±0.24
+27.16±32.41
+T5
+73.39±27.38
+51.93±18.99
+0.25±0.27
+inf
+T6
+79.53 ±17.18
+56.81±12.47
+0.20±0.18
+21.56±25.02
+T7
+71.12±29.38
+50.19±20.56
+0.27±0.30
+inf
+T8
+72.34±28.52
+48.93±19.57
+0.58±1.65
+35.37±29.53
+T9
+69.96±30.26
+48.75±19.66
+0.26±0.27
+inf
+T10
+69.28±28.39
+46.34±19.54
+0.36±0.44
+36.23±2.01
+T11
+52.87±29.66
+27.36±14.38
+2.16±4.86
+149.77±44.52
+T12
+64.76±31.42
+40.26±19.93
+0.52±0.76
+57.13±22.53
+T13
+67.16±33.19
+45.58±22.35
+0.27±0.29
+38.88±39.56
+Baseline [6]
+64.08±27.48
+46.21±20.12
+0.514±1.14
+277.63±163.00
+the official challenge website 6 for comprehensive introduction
+of their methods.
+C. Evaluation results and Analysis
+1) Segmentation performance: The segmentation perfor-
+mance of the baseline model and other participants’ algorithms
+for validation and testing set are illustrated in Table. III
+and Table. IV respectively. In Table. IV, we reported the
+average DSC, RVD, NSD and HD in the table, respectively.
+Our baseline model, SLEX-Net [6] obtained a DSC score of
+52.83%. Most of other teams improved the baseline model
+in all four metrics. The average DSC score, RVD, NSD for
+the participants lies in [40.22%,72.06%], [0.21, 1.55], and
+[25.11%, 53.59%], respectively. The best results on DSC,
+RVD, and NSD metrics achieved only 72.06%, 0.21, 53.59%,
+respectively. The overall performances are much lower than
+many other segmentation tasks, proving the great challenge of
+intracranial hemorrhage segmentation task. More importantly,
+most of the teams obtained ’infinite’ for the averaged HD
+because their method mistakenly diagnosed some difficult ICH
+6https://instance.grand-challenge.org/results/
+cases with tiny hemorrhages as normal subjects. The infinite
+results made it challenging to effectively rank the HD metric
+for different methods. In our challenge, we treat all ‘inf’ teams
+the same rank on HD which are inferior to others. Because we
+believe effectively diagnosis hard samples is also important
+in this task. Moreover, Fig. 2(a)-(d) demonstrate the results
+distribution across all the subjects in the testing dataset with
+box plots. It can be inferred that the standard deviations of the
+results distribution for top ranking teams are smaller that that
+of lower ranking ones, and also fewer outliers exists for them
+as well.
+2) Hematoma Volume Analysis: In this section, we ana-
+lyzed the relationship between hematoma volume size and
+the segmentation performances for different algorithms. The
+volume sizes of ICH are calculated by multiplying the
+voxel numbers of ICH and the pixel spacing in x,y,z di-
+mensions, which is consistent with the method in [6], [8].
+Fig. 3 highlights the correlation between volume size and
+the DSC scores with a scatter plot. It demonstrates that
+hemorrhages with small volume sizes are difficult to seg-
+ment, while large hematoma ICHs are relatively easier to
+achieve better segmentation results. Fig. 4 shows the segmen-
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+6
+TABLE IV: Summary of the INSTANCE 2022 testing phase. The average DSC, RVD, NSD and HD are reported for the
+baseline models and the submitted algorithms from each participant. The unit of HD is [mm]. The ranking is only provided
+for teams that successfully submitted the docker image and the technical paper descriptions in the testing phase. Bold values
+represent the best scores for each metric.
+Team
+DSC(%)↑
+NSD(%)↑
+RVD↓
+HD↓
+Ranking
+T1
+69.25±19.14
+53.59±15.65
+0.21±0.20
+35.27±28.47
+1
+T2
+72.06±21.07
+53.43±16.45
+0.26±0.25
+inf
+2
+T3
+69.00±24.68
+51.25±18.94
+0.31±0.28
+inf
+3
+T4
+68.94±25.06
+50.36±19.35
+0.32±0.29
+inf
+4
+T5
+67.97±25.07
+49.46±18.73
+0.32±0.28
+inf
+5
+T6
+67.39±26.91
+48.40±20.84
+0.32±0.31
+inf
+6
+T7
+66.84±24.75
+48.09±18.68
+0.33±0.27
+43.90±33.78
+7
+T8
+65.28±27.98
+47.49±21.70
+0.37±0.31
+inf
+8
+T9
+64.97±26.78
+46.86±19.58
+0.34±0.31
+inf
+9
+T10
+62.14±27.70
+42.01±19.88
+0.33±0.30
+inf
+10
+T11
+61.95±25.91
+42.80±18.75
+0.40±0.78
+55.36±26.53
+11
+T12
+57.04±28.20
+36.73±19.17
+0.43±0.50
+60.81±25.22
+12
+T13
+40.22±32.35
+25.11±21.54
+1.55±4.67
+68.36±41.79
+13
+Baseline [6]
+52.83±28.92
+38.42±21.04
+0.725±2.06
+309.06±287.31
+(a) Dice Coefficient
+(b) Normalized Surface Dice
+(c) Relative Volume Difference
+(d) Hausdorff Distance
+Fig. 2: Box plots of the experimental results on different evaluation metrics for all the submitted teams. The dots denote the
+individual scores of the 70 cases in the testing set.
+
+0.8
+DiceCoefficient(%
+0.6
+0.4
+0.2
+0.00.8
+SurfaceDice(%)
+0.6
+0.4
+0.2
+0.0
+T6T7T8T9Relative VolumeDifference
+1.00
+0.75
+0.50
+0.25
+0.00
+T1T2 T3 T4 T5 T6 T7 T8 T9 T10T11T12T13150
+HausdorffDistance
+100
+50
+T1 T2 T3 T4 T5 T6 T7 T8 T9 T10T11JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+7
+Fig. 3: The relationship between different Dice coefficients
+and the hematoma volume sizes demonstrates that the cases
+with smaller hematoma volumes are hard cases.
+Fig.
+4:
+The
+team-wise
+’Volumn-DSC’
+relationship
+fig-
+ure shows that the DSC scores improve with the in-
+crease of volume sizes for different algorithms from the
+participants. It is generated by separating the 70 test-
+ing cases with four different volume size groups: in-
+cluding [0, 4213], [4213, 7235], [7235, 19640], [19640, inf], re-
+spectively, and the average DSC score was calculated based
+on the results in each group.
+Fig. 5: The bar chart on Dice Coefficient for different kinds of
+intracranial hemorrhages shows that SAH is the most difficult
+class to segment.
+tation performance for all the methods with four hematoma
+volume size groups. It is generated by separating the 70
+testing cases with four different volume size groups: in-
+cluding [0, 4213], [4213, 7235], [7235, 19640], [19640, inf], re-
+spectively, and the average DSC score was calculated based on
+the results in each group. Fig. 4 further proves that the DSC
+scores improve with the increase of volume sizes for different
+algorithms from the participants.
+3) Hemorrhage Sub-type Analysis: Different sub-types of
+the intracranial hemorrhages are located at distinct positions
+of the brain, and patients can suffer from combinations of
+several kinds of hemorrhages. Certain types of hemorrhages
+usually present various different characteristics, leading to
+varied difficulties for distinguishing from normal brain tissues.
+Fig. 5 illustrates the average DSC value for different kinds
+of hemorrhages. It demonstrates that the SAH achieved the
+worst results in all metrics compared to other four kinds of
+ICHs. Hence, how to effectively segment SAH might be the
+most urgent problem needed to be solved to improve the ICH
+segmentation.
+D. Challenge Ranking Analysis
+Similar to the significance analysis in many biomedical
+image segmentation challenges [31], [32], we utilized the
+significance map to demonstrate the pairwise significant su-
+periority between different algorithms, as is illustrated in Fig.
+6. Specifically, we choose to perform significant test with one-
+sided Wilcoxon signed rank test at 5% significance level. In
+Fig. 6 (a-d), most of the yellow blocks are above the diagonal
+and the blue blocks are under the diagonal, indicating that most
+of the teams with smaller rank are significantly superior to
+those with larger ranks. Moreover, it also shows that different
+metrics have distinct ability to distinguish the good and bad
+
+1.0
+0.8
+0.6
+Dice
+0.4
+0.2
+0.0
+0
+30000
+00009
+00006
+120000 150000 180000
+Volumein[mm"T1
+0.8
+T2
+T3
+T5
+T6
+T7
+T8
+Average test DSC
+0.6
+T9
+T10
+T11
+T12
+T13
+0.4
+0.2
+0.0
+[0,4213]
+[4213,7235]
+[7235,19640]
+[19640,inf]
+Volume in [mm3]0.88
+DICE
+RVD
+0.8
+NSD
+0.73
+0.68
+0.67
+0.62
+0.63
+0.6
+0.52
+0.54
+0.41
+0.43
+0.4
+0.39
+0.36
+0.32
+0.29 
+0.2
+0.09
+0.0
+SDH
+EDH
+SAH
+IPH
+IVHJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+8
+(a) Significance map for Dice Coefficient
+(b) Significance map for Normalized Surface Dice
+(c) Significance map for Relative Volume Difference
+(d) Significance map for Hausdorff Distance
+Fig. 6: The significant superiority maps for ranking robustness analysis of different evaluation metrics. In each of the four
+maps, yellow blocks means that the evaluation metric for teams on the x-axis are significantly superior to those from the teams
+on the y-axis, which blue blocks means no significant superiority. The pairwise significant test with one-sided Wilcoxon signed
+rank test at 5% significance level is adopted in our experiment.
+performances among different algorithms. For example, the
+DSC, NSD and HD of T7 are significantly superior to that of
+T12, however, there exists no significant superiority on RVD
+metric.
+V. DISCUSSIONS
+A. 2D/3D architecture Choice
+The algorithm summary in section IV-B shows that the
+participants chose different algorithm implementations for 2D
+or 3D methods. We noticed that the winner method adopted
+the 2D/3D combination method, and most of the 3D methods
+outperformed the 2D implementations, yet we cannot draw
+definite conclusions on which kinds of methods are superior
+to another since there are many other factors contributing to
+the final results. However, we believe that directly utilizing 2D
+networks would lose significant context information among
+slices, which has been proved in numerous medical image
+segmentation tasks [6], [40]–[42]. Therefore, how to effec-
+tive incorporate inter-slice contextual information would be
+a fundamental problem for improving ICH segmentation. To
+this end, many participants utilized 3D UNet implementation,
+however, this might not be the optimal solution considering
+that the CT volumes in this challenge are anisotropic (pixel
+spacing: 0.42mm×0.42mm×5mm) [43], thus more effective
+techniques for exploiting inter-slice context for anisotropic
+volumes are needed.
+B. Bottlenecks for ICH segmentation
+The hematoma volume analysis in section IV-C2 demon-
+strates the inferior segmentation performance for hemorrhages
+
+T13
+T12
+TII
+T10
+T9
+T8
+T7
+T6
+T5
+T4
+T3
+T2
+T1
+T1
+T13T13
+T12
+T11
+T10
+T9
+T8
+T7
+T6
+T5
+T4
+T3
+T2
+T1
+T1
+T2T3T4T5T6T7T8T9T10T11T12T13T13
+T12
+T11
+T10
+T9
+T8
+T7
+T6
+T5
+T4
+T3
+T2
+T1
+T1
+T2T3T4T5
+T6T7T8T9T10 T11T12 T13T13
+T12
+T11
+T10
+T9
+T8
+T7
+T6
+T5
+T4
+T3
+T2
+T1
+T1
+T2JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+9
+with small volume sizes. The degradation of the segmentation
+indicates that the hemorrhage cases with small volume sizes
+are hard to segment. Fig. 3 shows that all the methods pro-
+posed by the participants have trouble dealing with very small
+hemorrhages. The majority of the cases that achieve a DSC
+score lower than 0.3 are those subjects with hemorrhage vol-
+ume smaller than 15000m3, and the overall DSC performances
+for all the subjects significantly deteriorate with substantial
+low DSC scores. Therefore, one important bottleneck for ICH
+segmentation is the small hemorrhage lesion segmentation, and
+effectively resolving this problem would certainty improve the
+overall segmentation performance and achieve better ranking
+in the challenge. Besides, the hemorrhage sub-type analysis in
+section IV-C3 shows that the subarachnoid hemorrhage (SAH)
+achieved the worst results in all metrics, with average DSC
+score for only 0.41. Thus, another bottleneck for ICH seg-
+mentation is how to deal with the subarachnoid hemorrhage.
+In conclusion, the future directions for the researches of ICH
+segmentation may be concentrated on the above-mentioned
+two bottlenecks. The researches of the hemorrhage diagnosis
+would be greatly improved by resolving these extremely hard
+cases.
+C. Evaluation Metrics Analysis
+We highly suggest the use of DSC, NSD and the RVD as
+the evaluation metrics for the ICH segmentation benchmark.
+According to the descriptions in section III-B, and section
+IV-C1. The HD and NSD are similar metrics that are used
+to evaluate the discrepancy between the target and predicted
+boundaries. However, we came across multiple extreme cases
+with average HD metrics equal to infinite when the predicted
+methods mistakenly diagnosed those hard cases with small
+hemorrhage lesions as normal head scans. The infinite values
+make it challenging to effectively rank different algorithms
+on that metric. However, the NSD metric has the same
+upper bound as DSC (100%), and there will be no such
+circumstances occur. Therefore, Hausdorff distance might not
+be a good metric choice for the INSTANCE challenge, and we
+consider abandoning it in the future INSTANCE challenges.
+D. Limitations and Future work
+Although this year’s INSTANCE challenge has achieved
+great success with numerous participants around the world, it
+still suffers from lots of limitations. They are mainly consist
+of three aspects:
+1) Data collection and annotation:
+Even though the
+INSTANCE2022 challenge has provided a relatively large
+dataset, they are mainly collected from a single institution
+with the same CT scanner. Although it could work in our
+challenge, it would definitely restrict the generalization of the
+model developed by different participants. In addition, for the
+data annotation, we only delineate the hemorrhage regions as
+foreground without considering the ICH sub-types, which are
+actually important information in clinical diagnosis and can
+also guide the segmentation of ICH.
+2) Task designs: In this years’ INSTANCE challenge, we
+only consider the hemorrhage segmentation task. However, it
+is also significant to perform ICH classification and hematoma
+volume quantification, which are highly clinical-related. The
+design of multiple tasks would simultaneously make the chal-
+lenge more comprehensive and provide more diverse research
+directions for the participants. In conclude, we will enhance
+the single-task challenge to a multi-task one in the future
+challenges.
+3) Source code Availability: In this years’ INSTANCE
+challenge, we highly recommended the participants to make
+their implementations to the public, and didn’t make it a
+mandatory option. As a result, we only find out one team make
+their code available. We didn’t demand them to share the code
+because we don’t expect it to be an obstacle for participating
+in this challenge. However, we notice that the code is too
+significant to be ignored for promoting the development in this
+research field. Therefore, we consider making it mandatory for
+top participants to make their code public available for future
+INSTANCE challenges.
+4) Future works for INSTANCE: We are currently working
+to promote the INSTANCE 2022 Challenge in many different
+aspects. Detailed improving directions are as follows:
+• More multi-institutional data. We will collect more ICH
+data from different CT scanner and different hospitals to
+improve the generalization of methods that are trained
+based on the INSTANCE benchmark.
+• More annotations and comprehensive task designs.
+We will annotate the different ICH sub-types of each CT
+scans and also calculate the hematoma volume of each
+cases to provide more clinical-related datasets. Mean-
+while, based on the above-mentioned extra annotations,
+we further expand the single-task challenge to a multi-
+task one, which simultaneously performs hemorrhage
+segmentation, classification and volume quantification
+tasks.
+• Mandatory options for open-source code. To pro-
+mote the advancement of the intracranial hemorrhage
+diagnosis, the top participants in the future INSTANCE
+challenge are required to share their code to the public.
+VI. CONCLUSION
+The INSTANCE challenge provides a novel benchmark
+for objectively measuring different intracranial hemorrhage
+segmentation methods in non-contrast head CT scans. A total
+of 13 teams successfully submitted their methods, and the
+winner solution achieved a DSC score of 0.6925 on the
+testing set, dramatically improving our baseline network. We
+have made the training set, the methodology descriptions and
+evaluation code public available on the challenge website,
+we hope this would promote the development in the ICH
+segmentation field. The challenge is now remains open for
+post-challenge submissions via Grand Challenge platform for
+benchmarking further algorithm exploitation. In the future,
+we will collect more multi-institutional data to improve the
+generalization of methods that are trained on the benchmark,
+and also perform more clinical-relevant annotations on ICH
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+10
+Fig. 7: Segmentation results for different ICH sub-types in terms of DSC and NSD scores. The blue color denotes the ICH
+lesion.
+sub-type and hematoma volumes and expand the single-task
+challenge to a multi-task one.
+ACKNOWLEDGMENTS
+We sincerely appreciate all the members in INSTANCE2022
+organization team for their hard work. Without your con-
+tinuous devotion to this challenge, it would not be that
+successful. This work was supported by the National Nat-
+ural Science Foundation of China under Grant 62001144,
+62272135 and 62001141, and by Science and Technology
+Innovation Committee of Shenzhen Municipality under Grant
+RCBS20210609103820029 and JCYJ20210324131800002.
+APPENDIX
+In (Li and Chen, 2022), Li and Chen used a combination
+of nnU-Net and uncertainty estimation ensemble strategy.
+Their experiments showed that even though the 2D nnU-
+Net could not achieve the overall dice accuracy of 3D nnU-
+Net, it performed better results than 3D nnU-Net when the
+intracranial hemorrhage had very small area or blurred bound-
+aries. Therefore, they use both 2D and 3D nnU-Net to predict
+the final result. Furthermore, in order to further alleviate the
+segmentation issue of small area intracranial hemorrhage and
+maintain stability during training, they utilized the weighted
+cross-entropy loss to replace simple cross-entropy loss in
+the nnU-Net. Due to the unbalanced intracranial hemorrhage
+types and intracranial hemorrhage areas, the models trained
+in different folds might predict completely different results.
+Simply average the predicted results from the models provide
+no additional benefit for these cases. To this end, they propose
+a simple but efficient uncertainty estimation ensemble strategy.
+
+DSC:89.9 NSD:65.4
+DSC:90.2 NSD:60.8
+DSC:90.4 NSD:60
+Case 1
+DSC:89.9 NSD:64.6
+DSC:93.3 NSD:69.9
+DSC:91.4 NSD:65.3
+Case 2
+DSC:55.3 NSD:46.3
+DSC:47.9 NSD:43.8
+DSC:48.4 NSD:39.8
+N
+Case 3
+DSC:65.1 NSD:45.9
+DSC:72.2 NSD:4S.6
+DSC:67.8 NSD:43.2
+Case 4
+DSC:67.7 NSD:48.9
+DSC:43.5 NSD:37.0
+DSC:29.1 NSD:19.7
+Case 5
+(a)Image
+(b)Ground Truth
+(c)T1
+(d)T3
+(e)T9JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+11
+For those cases with high uncertainty values, they use the
+voting method to get the final result. Use nnU-Net’s own data
+augmentation methods.
+In (Siddiquee et al., 2022 [44]), Siddiquee et al. used the
+2D version of encoder-decoder backbone based on with an
+asymmetrically larger encoder to extract image features and
+a smaller decoder to reconstruct the segmentation mask7. For
+the encoder part, they used 5 stages of down-sampling and
+2D ResNet blocks that each block’s output is followed by an
+additive identity skip connection. Furthermore, they used batch
+normalization and ReLU. For the decoder part, the decoder
+structure is similar to the encoder one, but with a single
+block per each spatial level. Each decoder level begins with
+upsizing with transposed convolution. In the preprocessing,
+they applied random rotation and random zoom on each axis
+with a probability of 0.4 and random contrast adjustment
+and random Gaussian noise with a probability of 0.2. The
+random coarse shuffle and random flips were applied on each
+axis with a probability of 0.5. In the training, they randomly
+split the entire dataset into 5-folds and trained a model for
+each. Moreover, they used L2 norm regularization on the
+convolution kernel parameters with a weight of 1e−5. The
+DiceCE loss is used for training.
+In (Sanner and Mukhopadhyay, 2022), Sanner and
+Mukhopadhyay used nnU-Net for the segmentation and pro-
+pose an evaluation of contour-based losses. Specifically, they
+integrated both the Hausdorff-distance loss as proposed by
+[45] and the contour loss proposed by [46]. While the former
+estimates the Hausdorff distance, the latter extracts the contour
+of both the prediction and the ground truth and minimizes the
+mean square error between them. In practice, Dice loss and
+CE loss were used as loss function and the Hausdorff-distance
+loss or the contour loss was used depending on the experiment.
+Furthermore, rather than using the standard z-normalization
+of nnU-Net for input images, they chose to clip the intensity
+values to [0 - 100]. A five-fold cross-validation was used to
+train five models and all models were ensembled to make the
+final prediction. The ”insane DA” scheme was used for data
+augmentation.
+In (Zhao et al., 2022), Zhao et al. used two stage 3D
+cascade U-Net network for ICH segmentation. For the stage 1,
+the basic module of the encoder and decoder is Conv-Instance
+Norm-LeakyReLU [47]. The operation of downsampling in
+the encoder is achieved by max pooling. The upsampling
+operation in the decoder is achieved by using the transpose
+convolution of 2 × 2 × 2. For the stage 2, a 3D U-Net was
+cascaded to the model, whose input is the output of probability
+map of the first stage. The 5-fold cross-validation was used for
+the training. In the preprocessing, the HU of CT images were
+clipped according to three different windows and levels, and
+corresponding range of HU were [0, 80], [-20, 180] and [-150,
+230]. The intensity of the voxel above the range were assigned
+the value of upper limit in range, and the intensity below the
+range is assigned the value of lower limit in range. Then the
+three images with different HU range clip were served as three
+channels and treated as one image.
+7Implementation: https://monai.io/apps/auto3dseg
+In (Zhang and Ma, 2022), Zhang and Ma used the
+standard nnU-Net.First, a 3D U-Net processes downsampled
+data, the resulting segmentation maps are upsampled to the
+original resolution.Then, these segmentations are concatenated
+as one-hot encodings to the full resolution data and refined
+by a second 3D U-Net. The preprocessing includes crop-
+ping,resampling and normalization. Meanwhile, random rota-
+tion, random scaling, random elastic transformation, gamma
+correction, and mirror were used to augment the data. The 3D
+nnU-Net was trained with an weighted combination of Dice
+loss and cross-entropy loss. The results on the test set were
+obtained as an ensemble of five models.
+In (Liu et al., 2022 [48]), Liu et al. used an ensemble
+model that combined viola-Unet and nnU-Net networks8.
+For the viola-Unet, it relies on voxels in feature space that
+intersect along orthogonal levels to provide an attention U-Net,
+which is an asymmetric encoder-decoder architecture with 7-
+depth layers. Overall, the Viola module is composed of three
+key blocks, i.e., the adaptive average pooling (AdaAvgPool)
+module that squeezes the input feature volume into three latent
+representation spaces along each axis of the input feature
+patch. The customized dense dilated convolutions merging
+(DDCM) networks fuses cross-channel and non-local contex-
+tual information on each orthogonal direction with adaptive
+kernel sizes, dilated ratios and strides. The Viola unit con-
+structs the voxels intersecting along orthogonal level attention
+volume based on fused and reshaped cross-channel-direction
+latent representation spaces. They trained all networks with
+randomly sampled patches of fixed size as input and applied
+a combination loss function of the dice loss and Focal loss
+for all their experiments. In preprocessing, CT image and
+ground truth labels were reoriented into ”RAS” format, then
+resized to a standard spacing of 1×1×5 mm3 using trilinear
+interpolation for the image and nearest-neighbor interpolation
+for the label. Each CT image was windowed into three image
+intensity ranges, and re-scaled to the range [0, 1] by min-max
+normalization and then stacked as 3-channel volumes to serve
+as inputs with the (C, H, W, D) shape, and then the 3-channel
+3D volume was normalized on only non-zero values with
+calculated mean and std on each channel separately. The data
+augmentations include random crop, random zoom, Gaussian
+noise, Gaussian smooth, rotation, random shift, random scale,
+flips, random contrast. Furthermore, they manually select
+the best prediction on each validation example from each
+submission as the pseudo-label and put them into our training
+set to fine-tune our models repeatedly in practice.
+In (Liang, 2022), Liang proposed a nnUNet-based method
+for 3-dimensional intracranial hemorrhage segmentation. In
+the preprocessing, the authors first deal with the data in method
+windowing and decide to choose a width of 59 and a center of
+96 for the image windowing by experiment. After windowing,
+in order to arrange the information of image, the author used
+a threshold to ensure the gray value of the image in a certain
+standard interval, unified data input. Then, downsampling the
+X and y axes, normalize the spacing of the slice axis to
+Slice down scale. A sampling includes maximum sampling,
+8Implementation: https://github.com/samleoqh/Viola-Unet
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+12
+average sampling, summation area sampling, and random area
+sampling. Finally, nnU-Net does the rest of the preprocessing.
+In the training, the author uses CE loss + DICE loss as
+loss function. Furthermore, to deal with category imbalance,
+oversampling was used, with 66.7% of the samples coming
+from random locations in the selected training sample’s, while
+33.3% of the patches were guaranteed to contain one of the
+foreground classes present in the selected training sample
+(randomly selected). The number of foreground patches was
+rounded to force a minimum value of 1 (resulting in one
+random patch and one foreground patch with a batch size of
+2). Use nnU-Net’s own data augmentation methods.
+In (Geiger et al., 2022), Geiger et al. used classic U-
+Net architecture and the network was conducted with the jax
+version of the e3nn library which enables the creation of neural
+networks equivariant to translations, rotations, and mirroring.
+Specially, the convolution kernels in the original architecture
+are replaced by a 3D e3nn voxel convolution of diameter
+5 mm. Furthermore, they used three 2x2x2 downsampling
+operations which halve the resolution in the encoding path
+and three corresponding trilinear upsampling operations on the
+decoding path. A Gaussian error linear unit activation function
+and instance normalization was used after each convolution.
+For the preprocessing, each CT volume was windowed to three
+different Hounsfield unit value ranges, scaled, and added to a
+separate channel which served as the model input. To increase
+the variety in the data, a random diffeomorphic deformation
+was performed on each training sample. The loss function em-
+ployed was cross-entropy loss. Eight models were trained, each
+on 80 randomly sampled subsets from the training dataset. The
+final prediction was performed by applying each of the eight
+models to patches of size 144x144x13 with padding discarding
+22x22x2 pixels on each side, a sliding window with an overlap
+of 26 pixels and Gaussian weighing, and then averaging the
+model outputs. For the final prediction, they take the ensemble
+average of the eight models.
+In (Qayyum et al., 2022), Qayyum et al. developed a
+coarse and fine segmentation model for intracranial hemor-
+rhage segmentations. They trained two different models for
+intracranial hemorrhage segmentations. In the first model, they
+trained 2DDensNet for coarse segmentation and cascaded the
+coarse segmentation masks output in the fine segmentation
+model along with input training samples. The proposed model
+is implemented made by a dense encoder followed by a non-
+dense decoder. The dense encoder consists of 5 dense blocks,
+each consisting of 6 dense layers followed by a transition
+layer. Each dense layer consists of 2 convolutional layers
+with batch normalization and ReLU activation functions. The
+model is trained using 5-fold cross-validation. To compute the
+final prediction, 2D images are stacked to make a 3D seg-
+mentation mask. The predicted segmentation mask is further
+cascaded in a fine segmentation model. In the fine stage, they
+used the nnU-Net model with fivefold cross-validation. The
+binary cross-entropy function was used as loss function. Hor-
+izontalFlip (p=0.5), VerticalFlip (p=0.5), and RandomGamma
+(p=0.8) were used to augment the dataset for training the
+proposed model. In addition, the dataset is normalized between
+0 and 1 using the max and min intensity normalization method.
+The training shape of each volume is fixed (256x256x16) and
+resample the prediction mask to the original shape for each
+validation volume using the linear interpolation method.
+In (Abramova et al., 2022), Abramova et al. used an
+approach based on a 3D U-Net architecture which incorporates
+squeeze-and-excitation blocks that similarly to their previous
+work [25]. For the preprocessing, coil removal and skull
+stripping were used, and a symmetric image was created
+for each case by flipping the original non-contrast CT and
+registering it to the initial one using the FLIRT algorithm from
+the FSL toolbox. For the normalization of input images, they
+performed percentile based range adjustment and used 0.5 and
+99.5 percentiles of brain-related voxels for clipping together
+with image-based calculated mean and standard deviation
+normalization. For the issue of class imbalance, they used
+a balanced sampling patch extraction technique, where we
+extracted an equal number of patches representing both classes
+from each image. Specifically, to avoid extracting a lot of
+patches from image background, they restricted the area to
+extract the negative patches within the brain mask and set a
+target number of patches to extract from each image in the
+training set. Half of them are uniformly extracted from the
+brain tissue area and represent negative class, while the other
+half is extracted from the lesion voxels. They augment the
+proposed dataset by choosing difficult cases and adding them
+into the training set again, meanwhile performing flipping and
+rotation, ensuring that more difficult patches are generated for
+training. The Dice loss and cross-entropy loss was used as
+loss function. To prevent overfitting, they used early stopping
+technique when approaching the minimal loss on validation
+set. The five-fold cross-validation strategy was used for the
+training. For the validation and testing stages, an ensemble
+with all the 5 models obtained in the cross-validation exper-
+iment was used to generate the final prediction masks. The
+probability masks obtained from the 5 models were averaged
+and thresholded to obtain the final binary mask for each case.
+Considering the results on the validation set, postprocessing
+was added to their pipeline to reduce the number of false
+positives. Specifically, as sizes of lesions vastly vary in the
+provided images, they remove all the lesions with the volume
+less than 10% of the biggest one in the post-processed image.
+In (Montagnon and Letourneau-Guillon, 2022), Mon-
+tagnon and Letourneau-Guillon used an ensemble approach
+including the Attention U-Net and SegResNet (with or without
+variational autoencoder) architectures combined with different
+loss functions. Specifically, they trained U-Net and SegResNet
+separately to use different loss functions including combina-
+tions of Dice with either Cross-Entropy loss or Focal loss,
+Tversky loss and Generalised Dice loss. Then leveraging all
+predictions, an ensemble voting approach allowed prediction
+of a final volume. Finally, to further remove potential false pos-
+itive predictions, predicted clusters were filtered by preserving
+ones with a volume larger than 36 pixels, an elevation above or
+equal to 3 slices and a mean density within [40; 80] HU range.
+In the preprocessing, in order to assess hemorrhage properties,
+they used DBSCAN, a density-based clustering algorithm, in
+order to extract connected pixels corresponding to hemorrhagic
+areas in each exam. Then they clipped images in the range [-
+
+JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2021
+13
+10; 140] HU. Taking into account the intracranial hemorrhage
+subtype distribution in the training dataset, they using Euler
+transforms consisting of rotations of either - π
+2 or - π
+2 around
+z-axis and translations ranging from -30 to 30 pixels, 10 pixels
+stepwise for subarachnoid and subdural hemorrhage subtypes
+images. Considering the limited size of the dataset, they used
+random orthogonal rotations and cropping for images in the
+training phase. In order to limit class imbalance issues, models
+were trained only on images containing at least one pixel of
+positive class. All models were trained using original images
+size (i.e. 512 × 512), clipped within [-10;140] and divided
+by the range of considered densities, which is 150 in their
+configuration.
+In (Roca et al., 2022), Roca et al. used a simple 2D Unet-
+like model and trained it with a binary cross-entropy loss.
+Especially, the model input is a layer that performs the clipping
+operation between [0, 256] and a normalization between [-
+0.5, 3.5] directly inside the model. In the preprocessing,
+they clipping the HU intensities in the soft tissue range of
+interest. For the data augmentation, they performed rotations
+and mirroring in the axial plane, plus some amount of intensity
+shift. Due to the data stratification was based on the presence
+of a segmentation on a given slice (positive cohort) vs. absence
+of segmentation (negative cohort), they used during training a
+balanced 50% / 50% of each cohort per mini-batch.
+In (Sindhura et al., 2022), Sindhura et al. proposed a deep
+learning framework which involves clinical knowledge and
+used U-Net3+ network for the segmentation. Specifically, they
+proposed a new data augmentation approach that leverages
+from the clinical knowledge that the two hemispheres of
+the human brain exhibit approximate symmetry. Due to the
+brain is approximately divided into two equal hemispheres by
+the midsagittal plane (MSP). So they use the MSP flipped
+versions of the CT scans as extra data. To extract MSP,
+they first apply the sobel edge detection method followed by
+thresholding to obtain the outline of the skull. An initial plane
+of reference is chosen to be the exact middle slice in the
+sagittal direction. A similarity metric is computed between the
+two hemispheres that are divided with the plane of reference.
+The reference plane is rotated by an angle of ±0.5◦. The
+plane which yields maximum similarity is the required MSP.
+Furthermore, to improve the robustness of the model, the
+usual data augmentations such as shear, rotation, zoom, flip,
+elastic transform, noise etc are being used. In view of there
+exists a very high class imbalance between the hematoma
+and non-hematoma pixels. So only the slices which contain
+hemorrhages are used in the training process and all slices
+of each scan are tested in the testing phase. In addition, to
+differentiate between the hemorrhage region and skull bone,
+which share similar intensities, they have performed skull
+stripping on each scan for both the training and testing process.
+The sum of focal loss and Dice similarity loss is used as the
+loss function in the training process.
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+arXiv:2301.03286v1  [eess.SP]  9 Jan 2023
+1
+A Dual-Function Radar-Communication System
+Empowered by Beyond Diagonal Reconfigurable
+Intelligent Surface
+Bowen Wang, Student Member, IEEE, Hongyu Li, Student Member, IEEE,
+Ziyang Cheng, Member, IEEE, Shanpu Shen, Member, IEEE,
+and Bruno Clerckx, Fellow, IEEE
+Abstract—This work focuses on the use of reconfigurable
+intelligent surface (RIS) in dual-function radar-communication
+(DFRC) systems to improve communication capacity and sensing
+precision, and enhance coverage for both functions. In contrast
+to most of the existing RIS aided DFRC works where the RIS
+is modeled as a diagonal phase shift matrix and can only reflect
+signals to half space, we propose a novel beyond diagonal RIS
+(BD-RIS) aided DFRC system. Specifically, the proposed BD-RIS
+supports the hybrid reflecting and transmitting mode, and is com-
+patible with flexible single/group/fully-connected architectures,
+enabling the system to realize full-space coverage. To achieve the
+expected benefits, we jointly optimize the transmit waveform, the
+BD-RIS coefficients, and sensing receive filters, by maximizing
+the minimum signal-to-clutter-plus-noise ratio for fair target
+detection, subject to the constraints of the communication quality
+of service, different BD-RIS architectures and power budget.
+To solve the non-convex and non-smooth max-min problem, a
+general solution based on the alternating direction method of
+multipliers is provided for all considered BD-RIS architectures.
+Numerical simulations validate the efficacy of the proposed
+algorithm and show the superiority of the BD-RIS aided DFRC
+system in terms of both communication and sensing compared
+to conventional RIS aided DFRC.
+Index Terms—Beyond diagonal reconfigurable intelligent sur-
+faces, dual-function radar-communication, full-space coverage,
+max-min optimization.
+I. INTRODUCTION
+In recent years, spectrum resources are becoming increas-
+ingly limited and valuable due to the exponential growth of
+services in wireless communications. Meanwhile, radar sys-
+tems are competing for the same scarce sources, which moti-
+vates the emergence of the dual-function radar-communication
+(DFRC) technology to achieve spectrum sharing between
+communication and radar. In DFRC systems, communication
+and radar functionalities are integrated on a common platform,
+which brings the benefit of enhanced spectrum efficiency while
+(Corresponding author: Ziyang Cheng, Shanpu Shen).
+B. Wang and Z. Cheng are with the School of Information & Communica-
+tion Engineering, University of Electronic Science and Technology of China,
+Chengdu, China. (email: B W [email protected], [email protected]).
+H. Li is with the Department of Electrical & Electronic Engineering, Impe-
+rial College London, London SW7 2AZ, U.K. (email: [email protected]).
+S. Shen is with the Department of Electronic and Computer Engineering,
+The Hong Kong University of Science and Technology, Clear Water Bay,
+Kowloon, Hong Kong (email: [email protected]).
+B. Clerckx is with the Department of Electrical & Electronic Engineering,
+Imperial College London, London, SW7 2AZ, U.K. and with Silicon Austria
+Labs (SAL), Graz A-8010, Austria (email: [email protected]).
+reducing power consumption and hardware costs. Therefore,
+DFRC is envisioned to play an important role in emerging
+environment-aware applications [1], such as vehicular net-
+works, environmental monitoring, and smart houses.
+Due to the benefits of DFRC, plenty of technical efforts have
+been devoted to designing DFRC systems. The design method-
+ology can be roughly divided into three categories: radar-
+centric design [2]–[4], communication-centric design [5]–[7],
+and joint waveform design [8], [9]. Radar-centric approaches
+utilize the radar waveform as the information carrier, where
+the communication symbols are embedded in conventional
+radar signals, such as linear frequency modulation [2] and
+frequency hopping [4]. On the other hand, communication-
+centric approaches realize the radar sensing tasks by modifying
+existing communication protocols [5] and waveforms [6], [7].
+In contrast to the first two categories [2]–[7], the DFRC
+waveforms can be jointly designed to provide more design
+freedoms so as to enhance both functionalities [8], [9]. Despite
+the above works [2]–[9] achieve satisfactory sensing and
+communication performance, one limitation is that they rely on
+the line-of-sight (LoS) links between the base station (BS) and
+communication users/sensing targets, which however yields
+the following two issues in practice: 1) The LoS link toward
+sensing targets or communication users can be easily blocked
+by obstacles. 2) The LoS channels may suffer from severe
+path loss especially for high frequencies.
+To overcome these issues, a promising technology named
+reconfigurable intelligent surface (RIS) [10]–[13] can be lever-
+aged. Specifically, RIS consists of numerous passive reconfig-
+urable scattering elements with low hardware cost and power
+consumption [10]–[13]. By properly placing and adjusting
+the RIS, it can establish virtual non-LOS (NLoS) links to
+“bypass” obstacles, and therefore compensate for the path
+loss and enhance system performance. Due to its advantages,
+RIS has been investigated for communications [14]–[16] and
+sensing [17]–[20] fields. Furthermore, RIS has been explored
+in various DFRC systems [21]–[27] to enhance both the
+communication and sensing performance, which are classified
+into the following two categories. The first category assumes
+LoS links exist from BS to users and targets. In this category,
+the RIS is used to compensate for the propagation loss and
+to improve the performance [21]–[23]. The second category
+focuses on the scenario where either communication users or
+sensing targets are blocked by barriers. In this category, RIS
+
+2
+is utilized to establish a NLoS link to bypass the barriers and
+thus enable DFRC [24]–[27].
+The limitation of the aforementioned works [21]–[27] is that
+they assume the RIS can only reflect signals towards the same
+side as the BS. In this case, both communication users and
+sensing targets should be located at the same side of RIS,
+i.e., within the same half-space, which limits the coverage
+and beam control flexibility of the RIS enabled DFRC sys-
+tem. To address this limitation, a novel hybrid transmissive
+and reflective RIS, namely simultaneously transmitting and
+reflecting RIS (STAR-RIS) [28] or intelligent omni-surface
+[29], is proposed to support signal reflection and transmission
+and thus extend the coverage. The integration of STAR-
+RIS and DFRC is first studied in [30], where the system is
+designed by minimizing the Cram´er-Rao bound (CRB) for
+radar target estimation subject to communication constraints.
+Then, a STAR-RIS is deployed at the vehicle to improve both
+sensing and communication performance [31]. Nevertheless,
+the achievable performance of STAR-RIS aided DFRC in [30],
+[31] is limited by the simple architecture of STAR-RIS without
+fully exploiting the architecture of RIS.
+To enhance the performance of RIS, a novel branch, namely
+beyond diagonal RIS (BD-RIS) [32]–[35], is proposed by
+exploring different architectures/modes of RIS. BD-RIS with
+group/fully-connected architectures under the reflective mode
+is first proposed in [32], which provides more controllable
+scattering matrices than conventional RIS. Then, the hybrid
+reflective and transmissive BD-RIS is proposed in [33] to
+achieve full-space coverage. It is proved that STAR-RIS is
+essentially a particular instance of two-port group-connected
+reconfigurable impedance network when each two antenna
+ports are connected to each other, namely cell-wise single-
+connected (CW-SC) architecture in [33]. More general cell-
+wise group/fully-connected (CW-GC/FC) architectures are
+also proposed based on the flexible connections among more
+antenna ports, which achieves better performance than STAR-
+RIS. Furthermore, a multi-sector BD-RIS is proposed in [35],
+which not only achieves full-space coverage but also provides
+higher performance gain than hybrid BD-RIS.
+Due to the benefits of BD-RIS, in this paper, we propose
+to adopt BD-RIS in DFRC systems to achieve full-space cov-
+erage and better performance. To the best of our knowledge,
+adopting BD-RIS in DFRC has not been investigated in the
+literature. In addition, in contrast to [30], [31] which ignore
+the signal-dependent clutters, we consider a more general and
+practical multi-target detection scenario with the presence of
+multiple clutters. The main contributions of this work are
+summarized as follows:
+Proposing BD-RIS aided DFRC. We propose a BD-RIS
+aided DFRC system, which consists of a BD-RIS enabling
+the full-space coverage, multiple users, and multiple sensing
+targets corrupted by multiple clutters. The BD-RIS divides
+the space into two sides and establishes virtual NLoS links
+for communication and sensing, where the dual-function BS
+(DFBS) performs communication tasks in one half space and
+sensing tasks in another side. To avoid multi-step path loss,
+we implement the radar sensing receiver on the BD-RIS for
+multi-target detection.
+Formulating Max-min fairness problem. We formulate
+the optimization problem to jointly design the transmit wave-
+form at the DFBS, the reflective and transmissive beamforming
+at the BD-RIS, and matched filters at the radar sensing
+receiver, to maximize the minimum radar output signal-to-
+clutter-plus-noise ratio (SCNR), subject to the communication
+quality of service (QoS) requirement for downlink communi-
+cations, the transmit power constraint at the DFBS, and the
+BD-RIS constraints with different architectures.
+Developing joint design framework. The joint design of
+BD-RIS aided DFRC is challenging due to the complicated
+and non-smooth objective, and newly introduced non-convex
+constraints of BD-RIS. To overcome these difficulties, we
+propose to decouple the BD-RIS constraints by the alternat-
+ing direction method of multipliers (ADMM) framework so
+that the resulting sub-problems are reformulated into easily
+handled forms and iteratively solved until convergence.
+Providing insights and numerical validation. We provide
+simulation results to illustrate the performance improvement
+achieved by BD-RIS. It is shown that benefiting from the
+high flexibility of BD-RIS, and the joint design of transmit
+waveform, BD-RIS, and the matched filters, the CW-GC/FC
+BD-RISs can achieve higher radar SCNR than CW-SC (STAR-
+RIS) ones under the same communication requirement. It
+is also shown the BD-RIS can substantially improve the
+performance and coverage compared to the conventional RIS,
+which shows the high flexibility of BD-RIS in manipulating
+the incident signal for enhancing the DFRC system.
+Organization: Section II presents the system model of
+the proposed BD-RIS aided DFRC. Section III formulates
+the max-min fairness problem and provides a joint design
+algorithm. Section IV evaluates the performance of the pro-
+posed algorithm and compares different BD-RIS architectures.
+Section V concludes this work.
+Notation: Scalars, vectors and matrices are denoted by stan-
+dard lowercase letter a, lower case boldface letter a and upper
+case boldface letter A, respectively. Cn and Cm×n denote
+the n-dimensional complex-valued vector space and m × n
+complex-valued matrix space, respectively. (·)T , (·)H, and
+(·)−1 denote the transpose, conjugate-transpose operations,
+and inversion, respectively. ℜ{·} and ℑ{·} denote the real and
+imaginary part of a complex number, respectively. ∥ · ∥F and
+| · | denote the Frobenius norm and magnitude, respectively.
+Diag(·) denotes a diagonal matrix. BlkDiag(·) denotes a block
+matrix such that the main-diagonal blocks are matrices and all
+off-diagonal blocks are zero matrices. IL indicates an L × L
+identity matrix.  denotes imaginary unit. ∠(·) represent the
+phase values of a matrix. Tr(·) denotes the summation of
+diagonal elements of a matrix. ⌊·⌋ is the round-down operation.
+II. SYSTEM MODEL
+As depicted in Fig. 1, we consider a DFRC system, where
+an NT-antenna DFBS simultaneously sends communication
+symbols to U single-antenna users and detects K targets in
+the presence of Q strong clutters with the assistance of an
+NS-cell BD-RIS. The BD-RIS adopts the hybrid transmissive
+and reflective mode, which divides the whole space into two
+
+3
+Target 
+Target 
+Clutter 
+Clutter 
+Transmissive Area 
+for Radar
+Cell 1
+BD-RIS
+Target 1
+Target K
+Clutter 1
+Clutter Q
+Reflective Area for 
+Communication
+Transmissive Area 
+for Radar
+User 
+User 
+DFBS
+Reflective Area for 
+Communication
+RIS elements
+Sensor elements
+User 1
+User NU
+DFBS
+NT
+Fig. 1. Illustration of a BD-RIS aided DFRC system.
+half areas, i.e., the transmissive and reflective areas. The DFBS
+provides communication services at the reflective area while
+performing radar sensing at the transmissive area aided by BD-
+RIS. The radar sensing receiver with NR antennas is installed
+adjacent to the BD-RIS to collect target echos and conduct
+target detection tasks. In the following subsections, we will
+review the modeling of BD-RIS with different architectures,
+and establish the communication and radar models.
+A. BD-RIS Architecture Model
+According to [33], the hybrid reflective and transmissive
+mode is essentially based on the group-connected reconfig-
+urable impedance network. Specifically, each two antenna
+ports are connected to each other, constructing one cell as
+illustrated in Fig. 1. Within each cell, two antennas with uni-
+directional radiation pattern are back to back placed such that
+each antenna covers half space. Mathematically, the BD-RIS
+with hybrid reflective and transmissive mode is characterized
+by two matrices, i.e., ΦR ∈ CNS×NS and ΦT ∈ CNS×NS.
+Depending on the inter-cell connection strategies, the BD-RIS
+can be categorized into the following three architectures.
+1) CW-SC BD-RIS Architecture: As shown in Fig. 2(a),
+we provide a simple example of CW-SC BD-RIS with 2 cells,
+from which we can observe that different RIS cells are not
+connected to each other. Therefore, matrices ΦT, ΦR are all
+restricted to be diagonal, i.e., ΦT = Diag(φT,1, . . . , φT,NS) and
+ΦR = Diag(φR,1, . . . , φR,NS), and satisfy
+|φT,i|2 + |φR,i|2 = 1, ∀i = 1, · · · , NS,
+(1)
+which conforms to the STAR-RIS constraints, indicating that
+the STAR-RIS is a special case of BD-RIS with CW-SC
+architecture [28], [29].
+2) CW-FC BD-RIS Architecture: Fig. 2(b) depicts an exam-
+ple of CW-FC BD-RIS with 2 cells. In contrast to CW-SC case,
+all cells of the CW-FC BD-RIS are connected to each other
+through reconfigurable impedance components. Accordingly,
+ΦT, ΦR are all full matrices satisfying
+ΦH
+T ΦT + ΦH
+R ΦR = INS.
+(2)
+3) CW-GC BD-RIS Architecture: As a balance between the
+above two extreme cases, CW-GC divides all cells into several
+Cell�
+User�
+User�
+BD�RIS
+DFBS
+Target�
+Target�
+Clutter�
+Clutter�
+Reflective�Area�for�
+Communication
+Transmissive�Area�
+for�Radar
+Antenna�3
+Antenna�1
+Antenna�4
+Z3,4
+Z3
+Z1,3
+Z1
+Z2,4
+Z1,2
+Z1,4
+Z2,3
+Z2
+Z4
+Antenna�2
+2�Cell�CW�FC�BD�RIS
+(b)
+Cell�1
+Cell�2
+Antenna�4
+Z2,4
+Z2
+Z4
+Antenna�2
+2�Cell�CW�SC�BD�RIS�
+(a)
+Cell�2
+Antenna�5
+Antenna�1
+Antenna�6
+Z5,6
+Z5
+Z1,5
+Z1
+Z2,6
+Z1,2
+Z1,6
+Z2,5
+Z2
+Z6
+Antenna�2
+4�Cell�CW�GC�BD�RIS
+Antenna�7
+Antenna�3
+Antenna�8
+Z7,8
+Z7
+Z3,7
+Z3
+Z4,8
+Z3,4
+Z3,8
+Z4,7
+Z4
+Z8
+Antenna�4
+Group�1
+Group�2
+(c)
+Cell�1
+Cell�2
+Cell�3
+Cell�4
+Antenna�3
+Antenna�1
+Z3
+Z1,3
+Z1
+Cell�1
+Fig. 2. Examples of (a) CW-SC BD-RIS, (b) CW-FC BD-RIS, and (c) CW-
+GC BD-RIS.
+groups and cells in each group adopt the the fully-connected
+architecture. Depending on the group division strategies, there
+are plenty of CW-GS architectures. For simplicity, here we
+consider the case where NS cells of the BD-RIS are uniformly
+divided into G groups and each group has the same size M =
+NS/G. For ease of understanding, an example of a 4-cell BD-
+RIS with CW-GC architecture having 2 groups is illustrated
+in Fig. 2(c). Hence, the model for CW-GC BD-RIS can be
+expressed as
+ΦT = BlkDiag(ΦT,1, . . . , ΦT,G),
+ΦR = BlkDiag(ΦR,1, . . . , ΦR,G),
+ΦH
+T,gΦT,g + ΦH
+R,gΦR,g = IM, ∀g = 1, · · · , G.
+(3)
+where ΦT,g ∈ CM×M and ΦR,g ∈ CM×M.
+Remark 1. The CW-GC architecture of BD-RIS is a general
+case, which becomes the CW-SC architecture (STAR-RIS) with
+a simple circuit when G = NS, and the CW-FC architecture
+achieving the best performance as G = 1. This means that
+CW-SC and CW-FC architectures are special cases of CW-GC
+architecture and the beam control flexibility/ability of CW-GC
+BD-RIS can be improved by decreasing G, but at the expense
+of increasing circuit complexity.
+B. Communication Model
+In this paper, we consider a standard multiuser multiple
+input single output (MISO) downlink scenario, where the
+DFBS provides communication services to the reflective area
+aided by the BD-RIS. We assume the direct links between the
+DFBS and downlink users are blocked and the channel state
+information (CSI) is available at the DFBS. The data symbol
+vector sl = [sl [1] , · · · , sl [U]]T ∈ CU contains the overall
+U data symbols in the l-th time slot, which are assumed to
+
+4
+be drawn from a standard M order phase-shift keying (M-
+PSK) modulation constellation. Furthermore, the data symbol
+vector sl is mapped to the transmit waveform w [l] ∈ CNT at
+the DFBS. Accordingly, the received signal of the u-th user
+at symbol time t is
+yu (t) = e2πfct
+L
+�
+l=1
+hH
+u ΦRGw [l] rect (t − l∆t) + nc,u (t) ,
+(4)
+where fc is the carrier frequency, L is the number of time slots
+during one transmission duration, G ∈ CNS×NT and hu ∈
+CNS stand for the channel coefficients of the communication
+links DFBS→BD-RIS and BD-RIS→u-th user, ∆t stands for
+symbol duration, rect (t) is the rectangle window function that
+takes the value 1 for t ∈ [0, ∆t] and 0 otherwise, and nc,u (t)
+is the additive white Gaussian noise (AWGN).
+By down converting the signal into baseband and sampling
+received signal yu (t) at the rate fs = 1/∆t within the symbol
+duration, the discrete baseband signal at the l-th time slot is
+yu [l] = hH
+u ΦRGw [l] + nc,u [l] ,
+(5)
+where nc,u [l] is the AWGN with zero mean and variance σ2
+C,u.
+In this work, we adopt the recently emerged symbol level
+beamforming (SLB) technology for communication in DFRC.
+Specifically, SLB technology utilizes the constructive inter-
+ference (CI), which is defined as the multi-user interference
+(MUI) that pushes the received symbols away from the detec-
+tion thresholds of the modulation constellation, to enhance the
+communication QoS while reducing BER [36], [37]. Here we
+briefly review the concept of SLB as follows.
+Fig. 3 takes quadrature-PSK (QPSK) as an example, where
+point A stands for the desired symbol sl [u] with the required
+signal-to-noise-ratio (SNR) threshold Γu,l of the u-th user, i.e.,
+−→
+OA =
+�
+σ2
+C,uΓu,lsl [u], and point D is the received noise-
+free signal, i.e., −→
+OD = ˜yu [l] = hH
+u ΦRGw [u]. The CI region
+refers to a polyhedron bounded by hyperplanes parallel to
+decision boundaries of the constellation, which is depicted
+as blue-shaded area in Fig. 3. The key of SLB is to enforce
+the received signal located in the CI region, which means the
+received signal is pushed away from decision boundaries and
+the SNR is guaranteed to be no less than the SNR threshold
+Γu,l. To mathematically depict the SLB constraint, we project
+point D into the direction of −→
+OA at point C, and extend −→
+CD
+to intersect with the nearest boundary of CI region at point B.
+Consequently, one of the criteria that specifies the location of
+−→
+OD in the CI region is
+|−→
+CD|
+|−→
+AC|
+=
+��ℑ
+�
+hH
+u ΦRGw [l] e∠(su[l])���
+ℜ
+�
+hH
+u ΦRGw [l] e∠(su[l])�
+−
+�
+σ2
+C,uΓu,l
+≤ tan Ω,
+(6)
+where Ω = π/M is half of the angular range of the CI resign.
+Remark 2. In this work, we adopt SLB instead of con-
+ventional block-level beamforing (BLB) due to the following
+two reasons: 1) By adopting SLB technology in our con-
+sidered DFRC system, we directly design transmit waveform
+W ∈ CNT×L for L time slots. However, the BLB in the
+D
+B
+� �
+�
+�
+u
+l
+y
+�
+�
+C
+Received
+Symbol
+� �
+�
+�
+2
+,
+u
+c
+u l
+l
+y
+� �
+�
+�
+�
+� �
+u l
+y�
+CI�Region
+2
+,
+c
+u l
+� �
+O
+�
+Imag
+Real
+A
+Fig. 3. Description of the CI region for a QPSK symbol.
+same scenario requires the design of the transmit beamformer
+Wl ∈ CNT×U, ∀l for all data symbols and time slots due
+to the linear mapping, which results in an increasing com-
+putational complexity [38]. 2) BLB regards the MUI as a
+harmful component and suppresses the MUI to guarantee
+communication QoS. However, the SLB utilizes the MUI to
+enhance the communication QoS, which provides additional
+design flexibility in DFRC [21].
+C. Radar Model
+To improve the sensing performance of the BD-RIS aided
+DFRC system, as shown in Fig. 1, we adopt a novel sensor-
+at-RIS architecture [20], where the radar receiving sensors
+are installed adjacent to the BD-RIS to collect the echo
+signals. This architecture greatly reduces the multi-step path-
+loss compared with the sensor-at-DFBS architecture [21]–[23].
+Moreover, we consider a scenario where the radar receiver
+attempts to detect K targets in the presence of Q strong
+clutters. Specifically, the k-th target of interest is characterized
+by angle ϕk and time delay τ k
+T , respectively, while the q-th
+clutter is characterized by angle ϑq and delay τq
+C, respectively1.
+The backscattered signal at the radar receiver after down
+conversion is thus [39]–[41]
+r (t) =
+K
+�
+k=1
+L
+�
+l=1
+αkA (ϕk) ΦTGw [l] rect
+�
+t − l∆t − τk
+T
+�
++
+Q
+�
+q=1
+L
+�
+l=1
+βqA (ϑq) ΦTGw [l] rect (t − l∆t − τq
+C)
++ nr (t) ,
+(7)
+where αk and βq, respectively, denote the propagation co-
+efficient for the k-th target and q-th clutter consisting of
+radar cross section (RCS) and channel propagation effects
+with E(|αk|2)
+=
+ζ2
+k
+and E(|βq|2)
+=
+ξ2
+q. A (ϕ)
+=
+aR (ϕ) aH
+T (ϕ) ∈ CNR×NS is the effective radar channel,
+where aT (ϕ) =
+1
+√NS [1, · · · , ej 2π
+λ d(NS−1) sin ϕ]T and aR (ϕ) =
+1
+√NR [1, · · · , ej 2π
+λ d(NR−1) sin ϕ]T denote the the transmit and
+1In this paper, we assume the targets and clutters are slowly moving or stay
+still, whose Doppler frequencies equal to zeros.
+
+5
+receive steering vector, respectively, with d and λ being
+element spacing and wavelength. nr (t) denotes AWGN.
+Then, we select the first target echo as the reference and
+sample the received signal r (t) at fs = 1/∆t, yielding the
+following received baseband signal
+R =
+K
+�
+k=1
+αkA (ϕk) ΦTGWJrk
+T
+�
+��
+�
+Target Echos
++
+Q
+�
+q=1
+βqA (ϑq) ΦTGWJrq
+C
+�
+��
+�
+Clutter Returns
++ Nr,
+(8)
+where Jr = [0L×r, IL, 0L×(Lobs−L−r)] ∈ CL×Lobs is the shift
+matrix with Lobs = L + {maxk rk
+T} − {mink rk
+T} being the
+receiver observation length, rk
+T = ⌊(τ k
+T − {min˜k τ ˜k
+T })fs⌋ the
+rang ring of the k-th target, and rq
+C = ⌊(τ q
+C − {mink τ k
+T })fs⌋
+the rang ring of the q-th clutter. Nr = [nr [1] , · · · , nr [L]] ∈
+CNR×L
+is
+the
+Gaussian
+noise
+matrix
+with
+nr [l]
+∼
+CN
+�
+0, σ2
+RINR
+�
+, ∀l.
+Finally, by performing the matched filter Uk ∈ CNR×Lobs
+to the k-th target at radar receiver, the k-th target detection
+problem can formulated as a binary hypothesis test [39]–[41]:
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Hk
+1 : αkUH
+k A (ϕk) ΦTGWJrk
+T
+(9a)
++
+K
+�
+p=1,p̸=k
+αpUH
+k A (ϕp) ΦTGWJrp
+T
++
+Q
+�
+q=1
+βqUH
+k A (ϑq) ΦTGWJrq
+C + Nr,
+(9b)
+Hk
+0 :
+K
+�
+p=1,p̸=k
+αpUH
+k A (ϕp) ΦTGWJrp
+T
+(9c)
++
+Q
+�
+q=1
+βqUH
+k A (ϑq) ΦTGWJrq
+C + Nr.
+(9d)
+According to the above binary hypothesis test (9), the detection
+probability P k
+D of the k-th target can be evaluated as [41]
+P k
+D = Q
+��
+2SCNRk,
+�
+−2 ln (Pfa)
+�
+,
+(10)
+where Q (·, ·) is the Marcum Q-function of order 1, Pfa is the
+false alarm probability, and the radar output SCNR of the k-th
+target after the matched filtering is given by
+SCNRk(W, ΦT, Uk) = ς−1
+k |Tr(αkUH
+k A(ϕk)ΦTGWJrk
+T )|
+2,
+(11)
+where ςk
+= �K
+p=1,p̸=k |Tr(αpUH
+p A (ϕp) ΦTGWJrp
+T )|
+2 +
+�Q
+q=1 |Tr(βqUHA (ϑq) ΦTGWJrq
+C)|
+2 + σ2
+R ∥Uk∥2
+F , ∀k.
+III. MAX-MIN FAIRNESS FOR BD-RIS AIDED DFRC
+In this section, we first formulate the joint design problem
+for BD-RIS aided DFRC, followed by a general algorithm.
+Finally, we propose an initialization scheme and analyze the
+computational complexity of the proposed algorithm.
+A. Problem Formulation
+Given that (10) is strictly increasing in SCNRk, for a
+specified value of false alarm probability Pfa, improving
+the detection probability P k
+D of the k-th target is equivalent
+to maximize the radar output SCNR of the k-th target.
+Moreover, for multiple target detection cases, beamforming
+design usually aims to improve the detection probability for
+all targets, especially for the weakest targets. Therefore, to
+improve the overall target detection probability and guarantee
+target detection fairness, we propose to maximize the minimal
+radar output SCNR among the K targets by jointly designing
+the transmit beamformer W, the BD-RIS matrices {ΦT, ΦR},
+and radar receiver filters {Uk}∀k, subject to communication
+QoS constraints, transmit power constraint, and BD-RIS con-
+straints. The joint design problem is thus formulated as2
+P1
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+max
+W,ΦT,ΦR,{Uk}
+�
+min
+∀k SCNRk (W, ΦT, Uk)
+�
+(12a)
+s.t.
+���ℑ
+�
+˜hH
+u w [l]
+����
+ℜ
+�
+˜hH
+u w [l]
+�
+−
+�
+σ2
+C,uΓu,l
+≤ tan Ω, (12b)
+∥W∥2
+F = E,
+(12c)
+ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) ,
+(12d)
+ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) ,
+(12e)
+ΦH
+T,gΦT,g + ΦH
+R,gΦR,g = ING, ∀g,
+(12f)
+where ˜hH
+u
+=
+hH
+u ΦRG is the equivalent channel for
+DFBS→DB-RIS→ u-th user and E is the transmit power.
+Problem P1 is a challenging non-convex problem. Partic-
+ularly, the non-convexity stems from the complicated frac-
+tional SCNR expression in the objective and highly coupled
+optimization variables. To simplify the joint design, in the
+following subsection, we propose a series of transformations
+and an ADMM based framework to decouple problem (12)
+into multiple more tractable sub-problems.
+B. Overview of Proposed Joint Design Framework
+To facilitate the joint design, we propose to re-arrange the
+SCNR (11) into explicit and compact forms. By defining uk =
+Vec(Uk), w = Vec(W) and φT = Vec (ΦT), and applying
+basic vectorization properties [42], the SCNR in (11) shares
+the following three equivalent expressions
+SCNRk (W, ΦT, Uk) =
+uH
+k ΨT,kuk
+uH
+k (ΨC,k + σ2
+RINRL) uk
+,
+(13a)
+=
+wHΥT,kw
+wHΥC,kw + σ2
+R ∥Uk∥2
+F
+,
+(13b)
+=
+φH
+T ΞT,kφT
+φH
+T ΞC,kφT + σ2
+R ∥Uk∥2
+F
+,
+(13c)
+where
+ΨT,k =ζ2
+k
+� ¯
+MT (k, ΦT)
+�
+wwH� ¯
+MT (k, ΦT)
+�H,
+2Based on the discussion in Remark 1, herein we focus on the design when
+the BD-RIS has CW-GC architecture, which is a general case including both
+CW-SC and CW-FC cases.
+
+6
+ΨC,k =
+K
+�
+p=1,p̸=k
+ζ2
+p
+� ¯
+MT (p, ΦT)
+�
+wwH� ¯
+MT (p, ΦT)
+�H
++
+Q
+�
+q=1
+ξ2
+q
+� ¯
+MC (q, ΦT)
+�
+wwH� ¯
+MC (q, ΦT)
+�H,
+ΥT,k =ζ2
+k
+� ¯
+MT (k, ΦT)
+�HukuH
+k
+� ¯
+MT (k, ΦT)
+�
+,
+ΥC,k =
+K
+�
+p=1,p̸=k
+ζ2
+p
+� ¯
+MT (p, ΦT)
+�HukuH
+k
+� ¯
+MT (p, ΦT)
+�
++
+Q
+�
+q=1
+ξ2
+q
+� ¯
+MC (q, ΦT)
+�HukuH
+k
+� ¯
+MC (q, ΦT)
+�
+,
+ΞT,k =ζ2
+k
+�
+˜
+MT (k, W)
+�H
+ukuH
+k
+�
+˜
+MT (k, W)
+�
+,
+ΞC,k =
+K
+�
+p=1,p̸=k
+ζ2
+p
+�
+˜
+MT (p, W)
+�H
+ukuH
+k
+�
+˜
+MT (p, W)
+�
++
+Q
+�
+q=1
+ξ2
+q
+�
+˜
+MC (q, W)
+�H
+ukuH
+k
+�
+˜
+MC (q, W)
+�
+,
+with ¯
+MT (k, ΦT) = JT
+rk
+T ⊗ (A (ϕk) ΦTG), ¯
+MC(q, ΦT) =
+JT
+rq
+C ⊗ (A (ϑq) ΦTG), ˜
+MT(k, W) = (JT
+rk
+T WT GT ) ⊗ A(ϕk),
+˜
+MC(q, W) = (JT
+rq
+CWT GT ) ⊗ A(ϑk).
+Based on the above derivations, the objective in problem P1
+is more tractable with respect to uk, w, or φT. However, it is
+still difficult to find the solution to P1 due to non-convex and
+coupled constraints (12b), (12c), and (12f). To tackle constraint
+(12f), we first define Φg = [ΦH
+T,g, ΦH
+R,g]H and rewrite (12f) as
+ΦH
+g Φg = IM. Then, we introduce auxiliary variables Θg =
+[ΘH
+T,g, ΘH
+R,g]H = Φg and decouple constraint (12f) into two
+separate constraints by adding the equality, which yields the
+following problem:
+P2
+
+
+
+
+
+
+
+
+
+
+
+
+
+max
+{Uk},W,{Φg},{Θg}
+�
+min
+∀k SCNRk (W, ΦT, Uk)
+�
+(15a)
+s.t.
+(12b), (12c), (12d), (12e),
+(15b)
+ΘH
+g Θg = IM, ∀g,
+(15c)
+Φg = Θg, ∀g.
+(15d)
+Problem P2 is a typical multi-variable optimization, which
+could be solved based on the ADMM framework using block
+coordinate descent (BCD) methods. To facilitate ADMM, we
+place the equality constraints Φg = Θg, ∀g into the objective
+function, and obtain the augmented Lagrangian (AL) as
+L ({Uk} , W, {Φg} , {Θg}) = −{min
+∀k SCNRk (W, ΦT, Uk)}
++
+G
+�
+g=1
+ℜ
+�
+Tr
+�
+ΛH
+g (Φg − Θg)
+��
++ ̺
+2
+G
+�
+g=1
+∥Φg − Θg∥2
+F ,
+(16)
+where Λg ∈ C2M×M, ∀g are dual variables associated with
+Φg = Θg, and ̺ ≥ 0 is the corresponding penalty parameter.
+Replacing the original objective function with AL function
+(16), we obtain the AL minimization problem as
+P2
+AL
+�
+min
+{Uk},W,{Φg},{Θg} L ({Uk} , W, {Φg} , {Θg}) (17a)
+s.t.
+(12b)-(12e), (15c).
+(17b)
+Now, the ADMM framework is constructed as follows, where
+the superscript of notations refers to the iteration index:
+Un+1
+k
+= arg min
+Uk L
+�
+{Uk} , Wn,
+�
+Φn
+g
+�
+,
+�
+Θn
+g
+��
+(18a)
+Wn+1 = arg min
+W L
+��
+Un+1
+k
+�
+, W,
+�
+Φn
+g
+�
+,
+�
+Θn
+g
+��
+s.t. (12b), (12c).
+(18b)
+�
+Φn+1
+g
+�
+= arg min
+Φg L
+��
+Un+1
+k
+�
+, Wn+1, {Φg} ,
+�
+Θn
+g
+��
+s.t. (12b), (12d), (12e).
+(18c)
+�
+Θn+1
+g
+�
+= arg min
+Θg L
+��
+Un+1
+k
+�
+, Wn+1,
+�
+Φn+1
+g
+�
+, {Θg}
+�
+s.t. (15c),
+(18d)
+Λn+1
+g
+= Λn
+g + ̺
+�
+Φn+1
+g
+− Θn+1
+g
+�
+.
+(18e)
+Variables (18a) to (18e) are successively updated by solving
+corresponding sub-problems until some stopping conditions
+are reached. In the following subsection3, we will elaborate
+on the solutions to sub-problems (18a) to (18d).
+C. Solution to Sub-problems
+1) Sub-problem w.r.t Uk: Given other variables, the opti-
+mization problem for updating Uk can be expressed as
+P2
+AL,{Uk}
+�
+max
+{Uk}
+�
+min
+∀k
+uH
+k ΨT,kuk
+uH
+k (ΨC,k + σ2
+RINRL) uk
+�
+.
+(19)
+It can be observed that P2
+AL,{Uk} is an unconstrained optimiza-
+tion problem and has K separable objective functions, each of
+which has the following form
+max
+Uk
+uH
+k ΨT,kuk
+uH
+k (ΨC,k + σ2
+RINRL) uk
+, ∀k.
+(20)
+Problem (20) is a classical generalized fractional quadratic
+optimization problem, whose optimal solution can be obtained
+by taking the generalized eigenvalue decomposition as [39]
+uk = EIG
+��
+ΨCN,k + σ2
+RINRL
+�−1 × ΨT,k
+�
+, ∀k.
+(21)
+where EIG (·) represents the eigenvector operator.
+2) Sub-problem w.r.t W: Given other variables, the opti-
+mization problem for updating W can be expressed as
+P2
+AL,W
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+max
+W
+�
+min
+∀k
+wHΥT,kw
+wHΥC,kw + σ2
+R ∥Uk∥2
+F
+�
+(22a)
+s.t.
+���ℑ
+�
+˜hH
+u w [l]
+����
+ℜ
+�
+˜hH
+u w [l]
+�
+−
+�
+σ2
+C,uΓu,l
+≤ tan Ω, (22b)
+∥W∥2
+F = E.
+(22c)
+Problem P2
+AL,W is hard to settle due to the non-smooth
+objective function and complicated non-convex constraints.
+To simplify the design, we first equivalently transform the
+3When introducing solutions to sub-problems, we omit the superscript of
+notations for conciseness unless otherwise stated.
+
+7
+objective into a smooth form by introducing an auxiliary
+variable γ, which yields the following problem
+P2−1
+AL,W
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+max
+W,γ
+γ
+(23a)
+s.t. min
+∀k
+wHΥT,kw
+wHΥC,kw + σ2
+R ∥Uk∥2
+F
+≥ γ,
+(23b)
+γ ≥ 0,
+(23c)
+(22b), (22c).
+(23d)
+Then, we deal with constraints (23b), (22b), and (22c) step-
+by-step detailed as follows.
+Step 1: Majorization minimization (MM) to (23b). We first
+rewrite constraint (23b) as
+wHΥC,kw − wHΥT,kw
+γ
++ σ2
+R ∥Uk∥2
+F ≤ 0, ∀k,
+(24)
+where the second term is a composite function with both
+w and γ. To simplify the joint design of problem (27), we
+perform MM and propose the following lemma.
+Lemma 1. Assume Υ is positive definite and γ > 0. A
+majorizer of f (w, γ) = wHΥw
+γ
+is
+f (w, γ; wn, γn) = 2ℜ
+�
+(wn)HΥw
+�
+γn
+− γ (wn)HΥwn
+(γn)2
+.
+Proof: Please refer to Appendix A.
+Using Lemma 1, we conduct the majorization on constraint
+(24) at point (wn, γn), yielding
+wHΥC,kw − 2ℜ
+�
+(wn)HΥT,kw
+�
+γn
++ γ (wn)HΥT,kwn
+(γn)2
++ σ2
+R ∥Uk∥2
+F ≤ 0, ∀k,
+(25)
+where γn is computed by
+γn = min
+∀k
+(wn)HΥT,kwn
+(wn)HΥC,kwn + σ2
+R ∥Uk∥2
+F
+.
+(26)
+Step 2: Reformulation to (22b). After some algebraic ma-
+nipulations, we rewrite (22b) as
+(22b) ⇔
+
+
+
+ℜ
+�¯hH
+u,1 (ΦR) w [l]
+�
+≥
+�
+σ2
+C,uΓu,l sin Ω,
+(27a)
+ℜ
+�¯hH
+u,2 (ΦR) w [l]
+�
+≥
+�
+σ2
+C,uΓu,l sin Ω,
+(27b)
+where ¯hu,1 (ΦR) = GHΦH
+R hu(sin Ω + e π
+2 cos Ω)e−∠(su[l])
+and ¯hu,2 (ΦR) = GHΦH
+R hu(sin Ω − e π
+2 cos Ω)e−∠(su[l]).
+Step 3: Simplification to (22c). We first scale the equality
+constraint (22c) as E−ǫ ≤ ∥W∥2
+F ≤ E+ǫ, where ǫ ≥ 0 is an
+auxiliary variable whose value approaches to zero. It is easy
+to notice that the right-hand side ∥W∥2
+F ≤ E + ǫ is convex,
+while the left-hand side E − ǫ ≤ ∥W∥2
+F is non-convex. To
+convexify the non-convex part, we perform MM and transform
+(22c) as two convex constraints
+�
+∥W∥2
+F − E − ǫ ≤ 0,
+(28a)
+2ℜ {Tr (WnW)} − ∥Wn∥2
+F − E + ǫ ≥ 0.
+(28b)
+Replacing non-convex contraints in (23) with (25), (27) and
+(28) based on Steps 1-3, and penalizing the slack variable ǫ
+into the objective function, we minimize the following problem
+P2−2
+AL,W
+
+
+
+
+
+min
+W,γ,ǫ −γ + κǫ
+(29a)
+s.t. γ ≥ 0, ǫ ≥ 0,
+(29b)
+(25), (27a), (27b), (28a), (28b) ,
+(29c)
+where κ ≥ 0 represents the penalty parameter to scale the
+impact of the penalty term. Problem P2−2
+AL,W is a convex
+second-order cone programming (SOCP) problem and can be
+globally solved by the interior point method (IPM).
+3) Sub-problem w.r.t {Φg}: Given other variables, the sub-
+problem for updating {Φg} is
+P2
+AL,{Φg}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+Φg −
+�
+min
+∀k
+φH
+T ΞT,kφT
+φH
+T ΞC,kφT + σ2
+R ∥Uk∥2
+F
+�
++
+G
+�
+g=1
+ℜ
+�
+Tr
+�
+ΛH
+g (Φg − Θg)
+��
++̺
+2
+G
+�
+g=1
+∥Φg − Θg∥2
+F
+(30a)
+s.t.
+���ℑ
+�
+˜hH
+u w [l]
+����
+ℜ
+�
+˜hH
+u w [l]
+�
+−
+�
+σ2
+C,uΓu,l
+≤ tan Ω, (30b)
+ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) ,
+(30c)
+ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) ,
+(30d)
+where ΦR and ΦT are separable in both objective and con-
+straints, and thus can be designed in parallel as follows.
+Solution to ΦR: The problem regarding ΦR is
+P2
+AL,ΦR
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+ΦR
+G
+�
+g=1
+ℜ
+�
+Tr
+�
+ΛH
+R,g (ΦR,g − ΘR,g)
+��
++̺
+2
+G
+�
+g=1
+∥ΦR,g − ΘR,g∥2
+F
+(31a)
+s.t.
+��ℑ
+�
+Tr
+� ¯Hu,lΦR
+����
+ℜ
+�
+Tr
+� ¯Hu,lΦR
+��
+−
+�
+σ2
+C,uΓu,l
+≤ tan Ω, (31b)
+ΦR = BlkDiag (ΦR,1, · · · , ΦR,G) ,
+(31c)
+where ΛR,g is extracted from the last M rows of Λg, ¯Hu,l =
+e∠(su[l])Gw [l] hH
+u . The difficulty of solving problem (31)
+comes from constraints (31b) and (31c), which can be tackled
+based on the following matrix arrangements. Specifically, we
+partition ¯Hu,l as
+¯Hu,l =
+
+
+¯H11
+u,l
+· · ·
+¯H1G
+u,l
+...
+...
+...
+¯HG1
+u,l
+· · ·
+¯HGG
+u,l
+
+ , ∀u, l,
+(32)
+where ¯Hij
+u,l ∈ CM×M. By defining ˜Hu,l = [ ¯H11
+u,l, · · · , ¯HGG
+u,l ],
+and re-arranging (31c) as ˜ΦR = [ΦR,1, · · · , ΦR,G], constraints
+(31b) and (31c) are merged into the following constraint
+���ℑ
+�
+Tr
+�
+˜Hu,l ˜ΦR
+�����
+ℜ
+�
+Tr
+�
+˜Hu,l ˜ΦR
+��
+−
+�
+σ2
+C,uΓu,l
+≤ tan Ω, ∀u, l,
+(33a)
+
+8
+⇔
+
+
+
+ℜ
+�
+Tr
+�
+ˆHu,l,1 ˜ΦR
+��
+≥
+�
+σ2
+C,uΓu,l sin Ω,
+ℜ
+�
+Tr
+�
+ˆHu,l,2 ˜ΦR
+��
+≥
+�
+σ2
+C,uΓu,l sin Ω,
+∀u, l, (33b)
+where ˆHu,l,1 =
+˜Hu,l
+�
+sin Ω + e− π
+2 cos Ω
+�
+and ˆHu,l,2 =
+˜Hu,l
+�
+sin Ω − e− π
+2 cos Ω
+�
+. This brings the following opti-
+mization problem
+P2−1
+AL,˜ΦR
+
+
+
+
+
+
+
+
+
+min
+˜ΦR
+ℜ
+�
+Tr
+�
+˜ΛH
+R
+�
+˜ΦR − ˜ΘR
+���
++̺
+2∥˜ΦR − ˜ΘR∥2
+F
+(34a)
+s.t.
+(33b),
+(34b)
+where
+˜ΘR
+=
+[ΘR,1, · · · , ΘR,G]
+and
+˜ΛR
+=
+[ΛH
+R,1, · · · , ΛH
+R,G]H. Problem P2−1
+AL,˜ΦR is a quadratic program
+(QP) with linear constraints and can be efficiently tackled
+via many existing optimization tools, such as the active set
+method and the primal-dual subgradient method [43].
+Solution to ΦT: The problem regarding ΦT is
+P2
+AL,ΦT
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+ΦT −
+�
+min
+∀k
+φH
+T ΞT,kφT
+φH
+T ΞC,kφT + σ2
+R ∥Uk∥2
+F
+�
++
+G
+�
+g=1
+ℜ
+�
+Tr
+�
+ΛH
+T,g (ΦT,g − ΘT,g)
+��
++̺
+2
+G
+�
+g=1
+∥ΦT,g − ΘT,g∥2
+F
+(35a)
+s.t. ΦT = BlkDiag (ΦT,1, · · · , ΦT,G) ,
+(35b)
+Similarly, we re-organize P2
+AL,ΦT into a concise form as
+P2−1
+AL,˜ΦT
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+˜ΦT,η
+−η + ℜ
+�
+Tr
+�
+˜ΛH
+T
+�
+˜ΦT − ˜ΘT
+���
++̺
+2
+���˜ΦT − ˜ΘT
+���
+2
+F
+(36a)
+s.t. min
+∀k
+˜φH
+T ˜ΞT,k ˜φT
+˜φH
+T ˜ΞC,k ˜φT + σ2
+R ∥Uk∥2
+F
+≥ η,
+(36b)
+η ≥ 0,
+(36c)
+where ˜ΦT = [ΦT,1, · · · , ΦT,G], ˜ΘT = [ΘT,1, · · · , ΘT,G],
+˜ΛT = [ΛH
+T,1, · · · , ΛH
+T,G]H with ΛT,g extracted from the first
+M rows of Λg, and ˜φT = Vec(˜ΦT). ˜ΞT,k = KGΞT,kKH
+G
+and ˜ΞC,k = KGΞC,kKH
+G, where KG = BlkDiag([IM ⊗
+[0M,(g−1)M, IM, 0M,(G−g)M]]G
+g=1) ∈ {0, 1}MNS×N 2
+S denotes
+the linear mapping matrix. Using Lemma 1 to simplify
+constraint (36b), we have
+P2−2
+AL,˜ΦT
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+˜ΦT,η
+−η + ℜ
+�
+Tr
+�
+˜ΛH
+T
+�
+˜ΦT − ˜ΘT
+���
++̺
+2
+���˜ΦT − ˜ΘT
+���
+2
+F
+(37a)
+s.t. ˜φH
+T ˜ΞC,k ˜φT −
+2ℜ
+�
+( ˜��n
+T)H ˜ΞT,k ˜φT
+�
+ηn
++η
+2ℜ
+�
+( ˜φn
+T)H ˜ΞT,k ˜φn
+T
+�
+(ηn)2
++σ2
+R ∥Uk∥2
+F ≤ 0, ∀k,
+(37b)
+η ≥ 0.
+(37c)
+Algorithm 1 Max-Min Fairness for BD-RIS Aided DFRC.
+Input: hu, ∀u, G, ̺ and system parameters.
+1: Initialize
+�
+U0
+k
+�
+, W0, Φ0
+T, and Φ0
+R.
+2: Set n = 1.
+3: repeat
+4:
+Calculate radar receive filters {Un
+k} by (21) in parallel.
+5:
+Update transmit waveform Wn by solving (29).
+6:
+Compute BD-RIS matrix Φn
+R by solving (34).
+7:
+Update BD-RIS matrix Φn
+T by solving (37).
+8:
+Obtain auxiliary variables
+�
+Θn
+g
+�
+by Theorem 1.
+9:
+Update dual variables
+�
+Λn
+g
+�
+by (18e).
+10:
+n = n + 1.
+11: until convergence.
+12: Return {Un
+k}, Wn, Φn
+T and Φn
+R.
+Output: {U⋆
+k} = {Un
+k}, W⋆ = Wn, Φ⋆
+T = Φn
+T, Φ⋆
+R = Φn
+R.
+Problem P2−2
+AL,˜ΦT is a convex SOCP and can be solved by IPM.
+4) Sub-problem w.r.t {Θg}: Given the other variables, the
+sub-problem for updating {Θg} is
+P2
+AL,{Θg}
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+min
+Φg
+G
+�
+g=1
+ℜ
+�
+Tr
+�
+ΛH
+g (Φg − Θg)
+��
++̺
+2
+G
+�
+g=1
+∥Φg − Θg∥2
+F
+(38a)
+s.t. ΘH
+g Θg = IM, ∀g,
+(38b)
+Problem P2
+AL,{Θg} can be split into G sub-problems, each of
+which has the following form
+P2−1
+AL,Θg
+
+
+
+
+
+
+
+
+
+min
+Φg ℜ
+�
+Tr
+�
+ΛH
+g (Φg − Θg)
+��
++̺
+2 ∥Φg − Θg∥2
+F
+(39a)
+s.t. ΘH
+g Θg = IM.
+(39b)
+Now, the remaining challenge of solving problem P2
+AL,Θg lies
+in the unitary constraint (39b). The unitary constraint (39b)
+forms a 2M dimensional complex Stiefel manifold [44], which
+can be approximately solved via manifold based algorithms,
+e.g., Riemannian conjugate gradient (RCG) and Riemannian
+trust regions (RTR). However, the iterative procedure of man-
+ifold methods might cause a lot of computational burdens.
+To speed-up the design, we provide a closed-form solution of
+problem P2
+AL,Θg in the following theorem.
+Theorem 1. With the unitary constraint (39b), the optimal
+solution for Θg is given by
+Θg = Bg [IM×M, 0M×M] DH
+g
+(40)
+where BgΣgDH
+g = Λg + ̺Φg is the singular value decom-
+position (SVD) of Λg + ̺Φg.
+Proof: Please refer to Appendix B.
+Based on the above derivations, the procedure of the above
+ADMM based algorithm is summarized in Algorithm 1.
+
+9
+D. Initialization Scheme
+Given that the ADMM procedure is usually sensitive to
+initial values, we present a 2-step initialization strategy to
+accelerate the convergence.
+Step1: Since it is not that straightforward to quickly find
+proper ΦT and ΦR, we randomly generate ΦT and ΦR, which
+satisfy the BD-RIS constraints.
+Step2: With initialized ΦR, we obtain the cascaded channel
+˜hH
+u (ΦR) = hH
+u ΦRG for the communication link. To provide
+a feasible and “good” initial point satisfying the constraint
+(12b), we initialize the transmit waveform W by solving the
+following QoS-constrained problem
+max
+W,Γ
+Γ
+s.t.
+ℜ
+�¯hH
+u,1 (ΦR) w [l]
+�
+≥
+�
+σ2
+C,uΓ sin Ω, ∀u, l,
+ℜ
+�¯hH
+u,2 (ΦR) w [l]
+�
+≥
+�
+σ2
+C,uΓ sin Ω, ∀u, l,
+∥W∥2
+F ≤ E,
+(41)
+which is a convex problem and can be efficiently solved by
+many numerical approaches [43].
+E. Complexity Analysis
+We provide a broad complexity analysis for Algorithms 1,
+which is summarized as follows
+1) Initialization: The main computational complexity of
+this stage comes from step 2 by solving the SOCP problem
+(41) with IPM, which requires approximately O
+�
+N 3
+TL3�
+.
+2) ADMM: This stage includes the iterative design of the
+radar receive filters Uk, transmit beamformer W, BD-RIS
+coefficients (ΦT, ΦR) and auxiliary variable {Θg}. Updating
+radar receive filters Uk requires O
+�
+KN 3
+R
+�
+. Solving problem
+(29) for updating W with IPM method needs complexity
+O
+�
+N 3
+TL3�
+. The complexity of updating BD-RIS coefficients
+(ΦT, ΦR) can be upper bounded by O
+�
+GN 3
+S
+�
+. Using Theo-
+rem 1 to update auxiliary variable {Θg} requires complexity
+of O
+�
+GM 3�
+. Therefore, the overall complexity of the ADMM
+framework is O(N0(KN 3
+R + N 3
+TL3 + GN 3
+S + GM 3)), where
+N0 denotes the maximum number of iterations.
+IV. PERFORMANCE EVALUATION
+In this section, we provided extensive simulation results to
+validate the effectiveness of the proposed algorithm and the
+performance of the proposed BD-RIS aided DFRC system.
+A. System Setup
+We assume that the DFBS equipped with NT = 8 antennas
+transmits QPSK symbols (M = 4) to U = 4 downlink users
+and detects K = 3 targets with the assistance of a BD-RIS
+having NS = 16 cells. The radar sensing receiver colocated
+with the BD-RIS has NR = 8 receive elements. The code
+length is L = 16 and the power budget at the DFBS is
+set as E = 10 W. The noise power at the users and radar
+sensing receiver are set as σ2
+C,u = σ2
+R = −100 dBm, ∀u.
+The communication QoS threshold is set the same for all
+users, i.e., Γu,l = Γ, ∀u, l. In addition, the distance-dependent
+TABLE I
+INFORMATION OF K TARGETS.
+Target Index
+Range (m)
+Azimuth (◦)
+RCS (dB)
+Target 1
+10
+30
+5
+Target 2
+14
+0
+8
+Target 3
+19
+-20
+10
+TABLE II
+INFORMATION OF Q CLUTTERS.
+No. of clutters
+Range (m)
+Azimuth (◦)
+RCS (dB)
+5
+15
+[20:2:28]
+25
+4
+20
+[-3:2:3]
+25
+9
+[6:1:14]
+10
+25
+5
+[16:1:20]
+-30
+25
+path loss is modeled as η (d) = ℵ (d/d0)−ℓ, where ℵ =
+−30 dB denotes the signal attenuation at the reference distance
+d0 = 1 m, and ℓ represents the path loss exponent. We
+set the path loss exponents for the DFBS→BD-RIS, BD-
+RIS→user, BD-RIS→target, and BD-RIS→clutter as 2.2, 2.2,
+2, and 2, respectively. The DFBS and BD-RIS are located as
+(−20 m, 0 m) and (0 m, 0 m), respectively, which results in
+the distance between DFBS and BD-RIS as dBR = 20 m.
+The U users are randomly located at reflective side with
+the same distance dRU = 16 m. The DFBS→BD-RIS and
+BD-RIS→user channels are assumed to follow the Rician
+fading model with the Rician factor being 3 dB. For the radar
+function, we assume K = 3 targets and 4 groups (Q = 23)
+of strong clutters are located in the transmissive side, whose
+detailed information is presented in Tables I and II. Moreover,
+we assume the range resolution as ∆d = 1 m, which indicates
+the radar sampling rate fs = 150 MHz. Combining Table I and
+the path loss model, the ratio of the propagation coefficients
+of the three radar targets is ζ2
+1 : ζ2
+2 : ζ2
+3 ≈ 3.2 : 1.6 : 0.7
+[17]–[19], [21], indicating that target 3 is the weakest target.
+B. Benchmark Schemes
+For comparison, we consider the following two benchmark
+schemes in the simulations.
+1) Benchmark 1: The radar-only case is selected as the up-
+per bound of the radar performance. We obtain this benchmark
+by changing the BD-RIS into transmissive mode and removing
+the downlink users, where the resultant problem can be tackled
+by modifying the proposed algorithm.
+2) Benchmark 2: We consider a doulbe-RIS case where
+one diagonal RIS working on the reflective mode while
+another
+working
+on
+the
+transmissive
+mode
+are
+adja-
+cently placed to achieve full-space coverage [30]. This
+baseline
+is
+a
+special
+case
+of
+BD-RIS
+with
+CW-SC
+where ΦT
+= Diag([φT,1, · · · , φT, NS
+2 ], 01× NS
+2 ) and ΦR
+=
+Diag(01× NS
+2 , [φR,1, · · · , φR, NS
+2 ]). Therefore, we can obtain this
+benchmark by modifying the proposed algorithm.
+C. Simulation Results
+1) Convergence Performance: In Fig. 4, we investigate
+the convergence of the proposed Algorithm 1 for different
+
+10
+0
+20
+40
+60
+80
+100
+120
+140
+160
+180
+200
+Number of Iteration
+0
+5
+10
+15
+20
+Radar Output SCNR (dB)
+CW-FC, Target 1
+CW-FC, Target 2
+CW-FC, Target 3
+CW-GC, Target 1
+CW-GC, Target 2
+CW-GC, Target 3
+CW-SC, Target 1
+CW-SC, Target 2
+CW-SC, Target 3
+(a)
+0
+50
+100
+150
+200
+250
+300
+Number of Iteration
+-5
+0
+5
+10
+15
+20
+Radar Output SCNR (dB)
+CW-FC, Target 1
+CW-FC, Target 2
+CW-FC, Target 3
+CW-GC, Target 1
+CW-GC, Target 2
+CW-GC, Target 3
+CW-SC, Target 1
+CW-SC, Target 2
+CW-SC, Target 3
+(b)
+Fig. 4. Radar output SCNR versus the number of iterations. (a) communica-
+tion threshold Γ = 0 dB, (b) communication threshold Γ = 15 dB.
+BD-RIS architectures. It can be observed that the proposed
+algorithm quickly converges to a stationary point. Specifically,
+after several iterations, all targets have nearly the same SCNR
+value, demonstrating that our algorithm can achieve fairness
+for multiple targets. Moreover, the CW-FC architecture enjoys
+faster convergence than other architectures under the same
+communication threshold. At the same time, the CW-SC re-
+quires nearly twice as many iterations of CW-FC to converge.
+For the same architecture, the proposed algorithm with a large
+communication threshold Γ needs more iterations to converge.
+This is due to the fact that if the intended communication
+threshold Γ is higher, fewer degrees of freedom (DoFs) in the
+optimization problem can be used.
+2) System Performance with Varying Parameters: In Fig.
+5, we study the minimum radar output SCNR versus the
+communication threshold Γ for different architectures. As ex-
+pected, the radar output SCNR monotonically decreases with
+Γ. This is because when the intended Γ is higher, less resource
+can be used to maximize the radar SCNR, which indicates
+that there is a trade-off between communication QoS and
+radar output SCNR. Meanwhile, the proposed algorithm with
+different architectures outperform the conventional RIS, which
+validates the advantage of deploying BD-RIS. In addition, the
+output SCNR gap between CW-FC/GC and CW-SC becomes
+large with increasing communication QoS requirement, which
+indicates that the advantage of CW-FC/GC BD-RIS is more
+prominent in high communication QoS requirement scenarios.
+Fig. 6 displays the minimum radar output SCNR as a
+0
+5
+10
+15
+20
+Communication QoS Threshold (dB)
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Minimum Radar Output SCNR (dB)
+Radar Only, CW-FC
+BD-RIS, CW-FC
+BD-RIS, CW-GC
+BD-RIS, CW-SC
+Double-RIS, CW-SC
+Fig. 5. Minimum radar output SCNR versus the communication threshold Γ
+for different architecture.
+10
+20
+30
+40
+50
+Transmit Power (W)
+5
+10
+15
+20
+25
+Minimum Radar Output SCNR (dB)
+Radar Only, CW-FC
+BD-RIS, CW-FC
+BD-RIS, CW-GC
+BD-RIS, CW-SC
+Double-RIS, CW-SC
+Fig. 6.
+Minimum radar output SCNR versus the transmit power E wit
+communication threshold Γ = 15 dB for different architectures.
+function of transmit power E under different architectures.
+It can be observed that the output SCNR for all schemes
+grows with the increase of transmit power E. Meanwhile,
+the growth of SCNR becomes slow when the transmit power
+is substantially large for all considered architectures. This
+is because we can improve transmit power to boost system
+performance to some degree, but excessive power will not
+improve performance further. Moreover, the slope variation of
+the BD-RIS scheme with CW-FC/GC/SC architectures is more
+significant than its competitors, indicating that CW-FC/GC/SC
+architectures are more sensitive to power budget.
+In Fig. 7, we present the minimum radar SCNR versus
+the number of groups G with different numbers of BD-RIS
+cells. We observe that with the same number of groups, the
+radar output SCNR increases with the increasing number of
+BD-RIS cells. The performance enhancement comes from
+the additional DoF of passive beamforming induced by the
+increasing number of cells, and the joint design of transmit
+waveform, the BD-RIS with more general constraints, and the
+
+11
+1
+2
+4
+8
+12
+16
+20 24
+32
+40
+Number of groups, G
+0
+2
+4
+6
+8
+10
+12
+14
+16
+18
+20
+Minimum Radar Output SCNR (dB)
+CW-SC
+CW-FC
+Fig. 7. Minimum radar output SCNR versus the number of groups G with
+different BD-RIS cells NS and communication threshold Γ = 15 dB.
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80
+Angle (Degree)
+-40
+-35
+-30
+-25
+-20
+-15
+-10
+-5
+0
+Normalized Transmit Beampattern (dB)
+Target 1
+Target 2
+Target 3
+Radar Only, CW-FC
+BD-RIS, CW-FC
+BD-RIS, CW-GC
+BD-RIS, CW-SC
+(a)
+-80
+-60
+-40
+-20
+0
+20
+40
+60
+80
+Angle (Degree)
+-40
+-35
+-30
+-25
+-20
+-15
+-10
+-5
+0
+Normalized Transmit Beampattern (dB)
+Target 1
+Target 2
+Target 3
+Radar Only, CW-FC
+BD-RIS, CW-FC
+BD-RIS, CW-GC
+BD-RIS, CW-SC
+(b)
+Fig. 8.
+Transmit beampattern of BD-RIS obtained via proposed algorithm
+for different architectures. (a) communication threshold Γ = 0 dB, (b)
+communication threshold Γ = 15 dB.
+matched filters, which also confirms the results in [33]. More
+importantly, the slope of each carve becomes steeper with the
+increasing number of groups, which indicates that the number
+of non-zero elements of BD-RIS matrices plays a significant
+role in increasing system performance.
+3) Radar Performance: In Fig. 8, we present the transmit
+beampattern obtained by the proposed algorithm. Results show
+that regardless of BD-RIS architectures, the transmit power
+(a) Radar-only, CW-FC
+(b) BD-RIS, CW-FD
+(c) BD-RIS, CW-GD
+(d) BD-RIS, CW-SD
+Fig. 9. The space-range beampattern behavors of the receive weights for the
+target 3 detection with communication threshold Γ = 10 dB.
+mainly concentrates around the three targets, which guarantees
+a high SCNR output at target directions. Moreover, the BD-
+RIS with CW-FC/GC architectures can focus more energy
+toward targets and has a lower sidelobe than that with CW-SC
+architecture, thanks to the more flexible passive beamfomring
+control provided by the CW-FC/GC architectures. We also
+observe that the transmit power towards target 3 is much high
+than other targets. This is because, as mentioned early, target
+3 is the weakest one, which needs more energy to improve
+the output radar SCNR. In addition, the transmit beampattern
+performance for BD-RIS with all architectures gets worse
+with larger communication QoS thresholds, which confirms
+the conclusion in Fig. 5.
+Fig. 9 shows the space-range beampattern of the designed
+waveform when BD-RIS has different architectures, where the
+beampattern of the k-th target is computed as P k
+R (θ, l) =
+|Tr{(U⋆
+k)H A (θ) ΦTGW⋆Jrl}|2 [39]–[41]. Without loss of
+generality, we take target 3 (k = 3) as an example to illustrate
+the space-range behavior of the designed waveform. Results
+show that the space-range beampattern can form a mainlobe
+at the location of the target k = 3 (green circle), but achieve
+null points at the locations of the other non-of-interest targets
+(red circles) and strong clutter sources (black rectangles) for
+all proposed architectures. This phenomenon can be explained
+as follows: i) To detect target k, the other targets are regarded
+as interference. ii) BD-RIS with more general architectures
+can provide more DoFs to resist strong clutters.
+V. CONCLUSION
+This paper considers the use of BD-RIS in the DFRC system
+in the presence of multiple targets and strong clutters. We
+start by reviewing the BD-RIS architectures, and deriving
+the communication and radar models. Our objective is to
+maximize the minimum radar output SCNR subject to the
+constraints of communication QoS, BD-RIS coefficients, and
+power budget. Then, a general algorithm utilizing the ADMM
+
+-30
+-40
+-50
+-600.8
+ncy (sino)
+0.6
+0.4-70
+-80
+-90
+-100
+-110
+120
+5
+30Normalized Spatial freque
+0.2
+0
+0.2
+0.4
+-0.6
+-0.8
+1
+5
+10
+15
+20
+Range (m)-30
+-40
+-50
+-600.8
+ncy (sino)
+0.6
+0.4-70
+-80
+-90
+-100
+-110
+120
+5
+30Normalized Spatial freque
+0.2
+0
+-0.2
+0.4
+0.6
+-0.8
+-1
+5
+10
+15
+20
+Range (m)-30
+-40
+-50
+-600.8
+ncy(sino)
+0.6
+0.4-70
+-80
+-90
+-100
+-110
+120
+5
+30Normalized Spatial freque
+0.2
+0
+-0.2
+0.4
+0.6
+-0.8
+-1
+5
+10
+15
+20
+Range (m)-30
+-40
+-50
+-600.8
+ncy (sino)
+0.6
+0.4-70
+dB
+-80
+-90
+-100
+-110
+-120
+5
+30Normalized Spatial freque
+0.2
+0
+0.2
+0.4
+0.6
+-0.8
+-1
+5
+10
+15
+20
+Range (m)12
+approach is developed to solve the resulting complicated non-
+convex max-min optimization problem. Finally, simulation
+results demonstrate the effectiveness of the proposed design
+algorithm, and the superiority of employing the BD-RIS in
+DFRC systems in terms of enhancing both communication and
+radar performance. Based on this initial work, there are many
+issues worth studying for future research on BD-RIS aided
+DFRC, such as wideband waveform design, the scenarios for
+target estimation, as well as exploring the application of multi-
+sector BD-RIS in DFRC systems.
+APPENDIX A
+PROOF OF LEMMA 1
+Given that f (w, γ) = wHΥw
+γ
+is jointly concave in w and
+γ when Υ ⪰ 0 and γ ≥ 0 [43], the first order approximation
+of f (w, γ), denoted by f (w, γ; wn, γn), is a majorizer of
+f (w, γ) at the point (wn, γn), which is
+f (w, γ; wn, γn)
+= f (wn, γn) + ( ∂f
+∂w|w=wn)T (w − wn)
+(42a)
++ ( ∂f
+∂w∗ |w=(wn)∗)T (w − (wn)∗)
++ (∂f
+∂γ |γ=γn)T (γ − γn) + ( ∂f
+∂γ∗ |γ=(γn)∗)T (γ − (γn)∗)
+= (wn)HΥwn
+γn
++ 2ℜ
+
+
+
+�
+2Υwn
+γn
+(wn)HΥwn
+(γn)2
+�H �
+w − wn
+γ − γn
+�
+
+
+= 2ℜ
+�
+(wn)HAw
+�
+γn
+− γ (wn)HΥwn
+(γn)2
+.
+The proof is thereby completed.
+APPENDIX B
+PROOF OF THEOREM 1
+We start by rewriting objective (39a) as [43]
+ℜ
+�
+Tr
+�
+ΛH
+g (Φg − Θg)
+��
++ ̺
+2 ∥Φg − Θg∥2
+F
+= −ℜ
+�
+Tr
+�
+ΘH
+g (Λg + ̺Φg)
+��
++ ̺
+2 ∥Φg∥2
+F + ̺M
+�
+��
+�
+constant
+.
+Then, problem P2−1
+AL,Θg can be symplified as
+max
+Φg
+ℜ
+�
+Tr
+�
+ΘH
+g (Λg + ̺Φg)
+��
+s.t. ΘH
+g Θg = IM.
+(43)
+Performing SVD to Λg + ̺Φg as BgΣgDH
+g = Λg +̺Φg, we
+can re-arrange the objective of (43) as
+ℜ
+�
+Tr
+�
+ΘH
+g (Λg + ̺Φg)
+��
+= ℜ {Tr (ΣgZg)} =
+M
+�
+i=1
+Σg [i, i] Zg [i, i] ,
+(44)
+where Zg = DH
+g ΘH
+g Bg. (44) achieves its maximum when
+Zg
+=
+IM×2M, yielding the optimal solution Θg
+=
+Bg [IM×M, 0M×M] DH
+g . The proof is thus completed.
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+

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+arXiv:2301.05066v1  [math.RT]  12 Jan 2023
+Branching symplectic monogenics using
+a Mickelsson–Zhelobenko algebra
+David Eelbode and Guner Muarem
+Abstract. In this paper we consider (polynomial) solution spaces for
+the symplectic Dirac operator (with a focus on 1-homogeneous solu-
+tions). This space forms an infinite-dimensional representation space for
+the symplectic Lie algebra sp(2m). Because so(m) ⊂ sp(2m), this leads
+to a branching problem which generalises the classical Fischer decom-
+position in harmonic analysis. Due to the infinite nature of the solution
+spaces for the symplectic Dirac operators, this is a non-trivial question:
+both the summands appearing in the decomposition and their explicit
+embedding factors will be determined in terms of a suitable Mickelsson-
+Zhelobenko algebra.
+Mathematics Subject Classification (2010). Primary 15A66, 17B10;
+Secondary 00A00.
+Keywords. Branching, Symplectic Dirac operator, Mickelsson–Zhelobenko
+algebra, simplicial harmonics.
+1. Introduction
+The Dirac operator is a first-order differential operator acting on spinor-
+valued functions which factorises the Laplace operator ∆ on Rm. It was
+originally introduced by Dirac in a famous attempt to factorise the wave op-
+erator, hence obtaining a relativistically invariant version of the Schr¨odinger
+equation. Since then, this operator has played a crucial role in mathemati-
+cal domains such as representation theory and Clifford analysis. The latter
+is a multidimensional function theory which is often described as a refine-
+ment of harmonic analysis, and a generalisation of complex analysis. It is
+centred around a generalisation of the operator introduced by Dirac (his
+operator /∂ is defined in 4 dimensions), and can be seen as a contraction
+between the generators ek for a Clifford algebra (acting as endomorphisms
+on so-called spinors) and corresponding partial derivatives ∂xk. To be more
+precise, introducing the Clifford algebra by means of the defining relations
+
+2
+David Eelbode and Guner Muarem
+{ea, eb} = eaeb + ebea = −2δab (with 1 ≤ a, b ≤ m) the Dirac operator is
+given by
+∂x =
+�
+e1
+. . .
+em
+�
+Idm
+
+
+
+x1
+...
+xm
+
+
+ =
+m
+�
+j=1
+ej∂xj ,
+whereby the (m × m)−identity matrix Idm has been added to explain what
+is meant by the ‘contraction’. Null-solutions for ∂x are called monogenics,
+and can be seen as generalisations of holomorphic functions. One often starts
+with the study of k-homogeneous polynomial solutions for the Dirac operator,
+which belong to the space Mk(Rm, S), where S stands for the aforementioned
+spinor space.
+An obvious generalisation of the operator ∂x can be obtained by using
+another matrix than Idm when contracting algebraic generators with partial
+derivatives. An important example is the symplectic Dirac operator, which
+is introduced on a symplectic space rather than an orthogonal space (see for
+example the work of Habermann [5]). This operator, denoted by Ds, is de-
+fined as a contraction between generators for a symplectic Clifford algebra
+and partial derivatives, using a skew-symmetric matrix Ω0 (rather than Idm).
+The symplectic Clifford algebra generators satisfy the Heisenberg relations
+[∂zj, zk] = δjk (the symplectic analogue of the Clifford relations for the gener-
+ators ek from above). Note that the symbols zj stand for real variables here,
+they are chosen because the sets of (real) variables xj and yj will also appear
+in this paper. In sharp contrast to the orthogonal case, the symplectic Clifford
+algebra is no longer finite-dimensional. This trend continues, in the sense that
+the associated symplectic spinor space S∞
+0 also becomes infinite-dimensional.
+In this paper, we study infinite-dimensional spaces defined in terms of
+solutions for the symplectic Dirac operator (generalised monogenics). These
+spaces can be defined algebraically
+S∞
+k = Ms
+k(R2m, S∞
+0 ) := Pk(R2m, C) ⊠ S∞
+0
+(k ∈ N).
+Here ⊠ denotes the Cartan product of the sp(2m)-representations Pk(R2m, C),
+the kth-symmetric power of the fundamental vector representation (modelled
+by polynomials), and the symplectic spinor space S∞
+0 (also referred to as the
+Segal-Shale-Weil representation). These spaces contain k-homogeneous S∞
+0 -
+valued solutions for the symplectic Dirac operator. The behaviour of these
+spaces as representations for sp(2m) is known (see e.g. [1] and the references
+therein), but in this paper we will look at these spaces as orthogonal repre-
+sentation spaces. This is motivated by the fact that so(m) ⊂ sp(2m), which
+means that we are dealing with a branching problem.
+In general, a branching problem can be described as follows: given a rep-
+resentation ρ of a Lie algebra g and a subalgebra h, we would like to under-
+stand how the representation ρ behaves as a h-representation. This restricted
+representation ρ|h will no longer be irreducible, but will decompose into h-
+irreducible representations. A branching rule then describes the irreducible
+
+Branching symplectic monogenics using a M–Z algebra
+3
+pieces which will occur, together with their multiplicities. For the symplec-
+tic spinors (i.e. for the space S∞
+0 ), this gives the Fischer decomposition in
+harmonic analysis, which means that the branching problem for S∞
+k leads to
+generalisations thereof. To describe the branching of the infinite-dimensional
+symplectic representation space S∞
+k
+under the inclusion so(m) ⊂ sp(2m),
+we will make use of a quadratic algebra which is known as a Mickelson-
+Zhelobenko algebra (see [9] for the general construction and properties).
+2. The symplectic Dirac operator and monogenics
+We will work with the symplectic space R2m and coordinates (x, y) equipped
+with the canonical symplectic form ω0 = �m
+j=1 dxj ∧ dyj. The matrix repre-
+sentation of the symplectic form is given by
+Ω0 =
+�
+0
+Idm
+−Idm
+0
+�
+.
+The group consisting of all invertible linear transformations preserving this
+non-degenerate skew-symmetric bilinear form is called the symplectic group
+and is formally defined as follows:
+Sp(2m, R) = {M ∈ GL(2m, R) | M T Ω0M = Ω0}.
+This is a non-compact group of dimension 2m2+m. Its (real) Lie algebra will
+be denoted by sp(2m, R). In the orthogonal case, the spin group determined
+by the sequence
+1 → Z2 → Spin(m) → SO(m) → 1
+plays a crucial role concerning the invariance of the Dirac operator ∂x and
+the definition of the spinors S. In the symplectic case, this role is played by
+the metaplectic group Mp(2m, R) fixed by the exact sequence
+1 → Z2 → Mp(2m, R) → Sp(2m, R) → 1.
+Despite the analogies, there are some fundamental differences:
+(i) First of all, the group SO(m) is compact, whereas Sp(2m, R) is not. This
+has important consequences for the representation theory. As a matter
+of fact, the metaplectic group is not a matrix group and does not admit
+(faithful) finite-dimensional representations.
+(ii) The orthogonal spinors S can be realised as a maximal left ideal in the
+Clifford algebra, but this is not the case for the symplectic spinors. The
+latter are often modelled as smooth vectors in the infinite-dimensional
+Segal-Shale-Weil representation (see [7] and the references therein). One
+can also identify the symplectic spinor space S∞
+0 with the space P(Rm, C)
+of polynomials in the variables (z1, . . . , zm) ∈ Rm, which is the approach
+we will use in this paper.
+Definition 2.1. Let (V, ω) be a symplectic vector space. The symplectic
+Clifford algebra Cls(V, ω) is defined as the quotient algebra of the tensor
+algebra T (V ) of V by the two-sided ideal Iω := {v ⊗ u − u ⊗ v + ω(v, u) :
+
+4
+David Eelbode and Guner Muarem
+u, v ∈ V }. In other words Cls(V, ω) := T (V )/Iω is the algebra generated by
+V in terms of the relation [v, u] = −ω(v, u), where we have omitted the tensor
+product symbols.
+Definition 2.2. Denote by ⟨u, v⟩ := �m
+k=1 ukvk the canonical inner product
+on Rm (where we allow partial derivatives to appear as coefficients, see the
+operators below). We then define the following operators acting on polyno-
+mial functions in P(R3m, C):
+(i) The symplectic Dirac operator Ds = ⟨z, ∂y⟩ − ⟨∂x, ∂z⟩.
+(ii) The adjoint operator Xs = ⟨y, ∂z⟩+⟨x, z⟩ with respect to the symplectic
+Fischer product (see Section 5 of [2] for more details).
+(iii) The Euler operator E = �m
+j=1(xj∂xj + yj∂yj) = Ex + Ey measuring the
+degree of homogeneity in the base variables (x, y) ∈ R2m.
+Note that some authors use the notation ⟨∇x, ∇y⟩ for an expression such as
+�
+k ∂xk∂yk, but we will use the Dirac operator symbol here instead of the
+nabla operator.
+Lemma 2.3. The three operators X =
+√
+2Ds, Y =
+√
+2Xs and their commu-
+tator H = [X, Y ] = −2(Ex + Ey + m) give rise to a copy of the Lie algebra
+sl(2).
+One now easily sees that the symplectic Dirac operator is nothing more than
+the contraction between the Weyl algebra generators (zk, ∂zk) with the vector
+fields (∂xk, ∂yk) for k = 1, . . . , m using the canonical symplectic form Ω0.
+Definition 2.4. The space of k-homogeneous symplectic monogenics is de-
+fined by S∞
+k := ker(Ds)∩
+�
+Pk(R2m, C) ⊗ P(Rm, C)
+�
+, where the space P(Rm, C)
+in the vector variable z ∈ Rm plays the role of the symplectic spinor space
+S∞
+0 .
+Note that as an sp(2m, R)-module, S∞
+k is reducible and decomposes into two
+irreducible parts: S∞
+k = S∞
+k,+ ⊕ S∞
+k,− with highest weights
+S∞
+k,+ ←→
+�
+k − 1
+2, −1
+2, . . . , −1
+2
+�
+and
+S∞
+k,+ ←→
+�
+k − 1
+2, −1
+2, . . . , −3
+2
+�
+.
+These weight entries are fixed by the Cartan algebra h = Alg(Xjj : 1 ≤ j ≤
+m), where the elements Xjj are defined in the lemma below. In this paper, we
+will omit the parity signs and work with S∞
+k as a notation which incorporates
+both the positive and negative spinors (in our model, this will correspond to
+even or odd in the variable z ∈ Rm, see below, so it is always easy to ��decom-
+pose’ into irreducible components when necessary).
+The three operators from Lemma 2.3 can be proven to be invariant under the
+action of the symplectic Lie algebra, in the sense that they commute with
+the following generators (see also Lemma 3.3 in [3]):
+
+Branching symplectic monogenics using a M–Z algebra
+5
+Lemma 2.5. The symplectic Lie algebra sp(2m) has the following realisation
+on the space of symplectic spinor-valued polynomials P(R2m, C) ⊗ S∞
+0 :
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+Xjk = xj∂xk − yk∂yj − (zk∂zj + 1
+2δjk)
+1 ≤ j, k ≤ m
+Yjk = xj∂yk + xk∂yj − ∂zj∂zk
+1 ≤ j < k ≤ m
+Zjk = yj∂xk + yk∂xj + zjzk
+1 ≤ j < k ≤ m
+Yjj = xj∂yj − 1
+2∂2
+zj
+1 ≤ j ≤ m
+Zjj = yj∂xj + 1
+2z2
+j
+1 ≤ j ≤ m
+(2.1)
+The branching rule for S∞
+0 , when considering it as a representation space for
+the orthogonal Lie algebra so(m) ⊂ sp(2m), leads to the Fischer decomposi-
+tion for C-valued polynomials in the variable z ∈ Rm (see below). Note that
+so(m) is generated by the operators Xjk −Xkj for 1 ≤ j < k ≤ m, giving rise
+to the well-known angular operators ubiquitous in quantum mechanics (often
+denoted by Lab with 1 ≤ a < b ≤ m). In our previous paper [3], we therefore
+tackled the next case k = 1 as this is a natural generalisation of said Fischer
+decomposition. The main problem with our branching rule (Theorem 5.6 in
+[3]) is the fact that these so(m)-spaces appear with infinite multiplicities,
+which are not always easy to keep track of. Therefore the main goal of this
+paper is to show that one can organise these in an algebraic framework which
+extends to other values for k too, using a certain quadratic algebra.
+3. Simplicial harmonics in three vector variables
+In this section we describe a generalisation of harmonic polynomials, in three
+vector variables. This will be done in terms of a solution space for a ‘natural’
+collection of so(m)-invariant differential operators. The corresponding Howe
+dual pair will be useful for the branching problem addressed above. For the
+sake of completeness, we recall the following basic definition:
+Definition 3.1. A function f(x) on Rm is called harmonic if ∆f(x) = 0. The
+k-homogeneous harmonics are defined as Hk(Rm, C) := Pk(Rm, C) ∩ ker(∆).
+These spaces define irreducible representations for so(m) with highest weight
+(k, 0, . . . , 0) for all k ∈ Z+.
+It is well-known that the space of k-homogeneous polynomials Pk(Rm, C) is
+reducible as an so(m)-module (see for example [4]) and decomposes into har-
+monic polynomials. In fact, the decomposition of the full space of polynomials
+is known as the aforementioned Fischer decomposition, given by
+P(Rm, C) =
+∞
+�
+k=0
+Pk(Rm, C) =
+∞
+�
+k=0
+∞
+�
+p=0
+|z|2pHk(Rm, C).
+This can all be generalised to the case of several vector variables (sometimes
+also called ‘a matrix variable’): for any highest weight for so(m) there is a
+(polynomial) model in terms of simplicial harmonics (or monogenics for the
+half-integer representations). We refer to [8] for more details. In this paper,
+
+6
+David Eelbode and Guner Muarem
+we will consider these spaces for so(m)-weights characterised by three inte-
+gers (a, b, c) where a ≥ b ≥ c ≥ 0. Also note that trailing zeros in the weight
+notation will be omitted from now on, so for instance (k, 0, . . . , 0) will be writ-
+ten as (k). First of all, we consider homogeneous polynomials Pa,b,c(z; x, y)
+in three vector variables (z; x, y) ∈ R3m. Here we use the notation (z; x, y)
+to stress the difference between the variable z (the spinor variable, refer-
+ring to an element in S∞
+0 ) from the other two variables (x, y) ∈ R2m, which
+are ‘ordinary’ variables. The parameters (a, b, c) then refer to the degrees of
+homogeneity in (z; x, y). These polynomials carry the regular representation
+of the orthogonal group (or the derived so(m)-action in terms of angular
+momentum operators Lab from above).
+We further introduce the Weyl algebra in three vector variables as the
+algebra generated by the variables and their corresponding derivatives:
+W(R3m, C) := Alg(xα, yβ, zγ, ∂xδ, ∂yε, ∂zζ) with α, β, γ, δ, ε, ζ ∈ {1, . . ., m} .
+Just like in the case of the classical Fischer decomposition, where the Lie
+algebra sl(2) appears as a Howe dual partner, there is a Lie algebra appearing
+here. To be precise, it is the Lie algebra sp(6) = g−2⊕g0⊕g+2, with parabolic
+subalgebra p := g−2 ⊕ g0 and Levi subalgebra g0 ∼= gl(3). The subspaces g±2
+contain six ‘pure’ operators each (i.e. only variables, acting as a multiplication
+operator, or only derivatives). More specifically, the subspaces are spanned
+by the following SO(m)-invariant operators:
+g−2 := span(∆x, ∆y, ∆z, ⟨∂x, ∂y⟩, ⟨∂y, ∂z⟩, ⟨∂x, ∂z⟩)
+g0 := span(⟨x, ∂y⟩, ⟨y, ∂x⟩, ⟨x, ∂z⟩, ⟨z, ∂x⟩, ⟨y, ∂z⟩, ⟨z, ∂y⟩, Ex, Ey, Ez)
+g+2 := span(|x|2, |y|2, |z|2, ⟨x, y⟩, ⟨y, z⟩, ⟨x, z⟩)
+Definition 3.2. The space of Howe harmonics of degree (a, b, c) in the vari-
+ables (z, x, y) is defined as H∗
+a,b,c(R3m, C) := Pa,b,c(R3m, C) ∩ ker(g−2).
+In what follows the notation ker(A1, . . . , An) stands for ker(A1)∩. . .∩ker(An),
+so ker(g−2) means that simplicial harmonics are annihilated by all (pure dif-
+ferential) operators in sp(6). As a representation space for so(m), the spaces
+H∗
+a,b,c are not irreducible. In order to obtain an irreducible (sub)space, we
+have to impose extra conditions.
+Definition 3.3. The vector space of simplicial harmonics of degree (a, b, c)
+in the variables (z, x, y) is defined by means of
+Ha,b,c(R3m, C) := H∗
+a,b,c(R3m, C) ∩ ker
+�
+⟨z, ∂x⟩, ⟨z, ∂y⟩, ⟨x, ∂y⟩
+�
+.
+As was shown in [8], this defines an irreducible representation space for so(m)
+with highest weight (a, b, c), where the dominant weight condition a ≥ b ≥ c
+must hold. This now leads to the following generalisation of the result above
+(the Fisher decompostion in three vector variables):
+
+Branching symplectic monogenics using a M–Z algebra
+7
+Theorem 3.4. The space P(R3m, C) of complex-valued polynomials in three
+vector variables (in Rm) has a multiplicity-free decomposition under the ac-
+tion of sp(6) × SO(m) by means of:
+P(R3m, C) ∼=
+�
+a≥b≥c
+V∞
+a,b,c ⊗ Ha,b,c(R3m, C),
+where we used the dominant weight condition in the summation. The notation
+V∞
+a,b,c hereby refers to a Verma module (see for example [6]) for sp(6).
+4. The Mickelsson-Zhelobenko algebra (general setup)
+We have now introduced 21 differential operators giving rise to a realisation
+of the Lie algebra sp(6) inside the Weyl algebra (on 3 vector variables in
+Rm). In this section we construct a related algebra, the so-called Mickelsson-
+Zhelobenko algebra (also called transvector or step algebra) Z. Let g be
+a Lie algebra and let s ⊂ g be a reductive subalgebra. We then have the
+decomposition g = s ⊕ t, where t carries an s-action for the commutator (i.e.
+[s, t] ⊂ t). For s we then fix a triangular decomposition s = s− ⊕ h ⊕ s+,
+where s± consists of the positive (resp. negative roots) with respect to the
+Cartan subalgebra h ⊂ s. We then also define a left ideal J ⊂ U(g) in the
+universal enveloping algebra U(g) by means of U(g)s+. This allows us to
+define a certain subalgebra of U(g) which is known as the normaliser:
+Norm(J) := {u ∈ U(g) | Ju ⊂ J}.
+The crucial point is that J is a two-sided ideal of Norm(J), which allows us
+two define the quotient algebra S(g, s) = Norm(J)/J which is known as the
+Mickelsson algebra.
+In a last step of the construction, we consider an extension of U(g) to a
+suitable localisation U′(g) given by
+U′(g) = U′(g) ⊗U(h) Frac(U(h)) ,
+where Frac(U(h)) is the field of fractions in the (universal enveloping algebra
+of the) Cartan algebra. The ideal J′ can be introduced for this extension
+too (in a completely similar way) and the corresponding quotient algebra
+Z(g, s) := Norm(J′)/J′ is the Mickelsson-Zhelobenko algebra. These two al-
+gebras are naturally identified, since one has that
+Z(g, s) = S(g, s) ⊗U(h) Frac(U(h)) .
+Note that this algebra is sometimes referred to as a ‘transvector algebra’,
+which is what we will often use in what follows.
+5. The Mickelsson-Zhelobenko algebra Z(sp(6), so(4))
+We will now define a specific example of the construction from above, which
+will help us to understand how the branching of S∞
+k
+works. First of all, we
+note the following:
+
+8
+David Eelbode and Guner Muarem
+Lemma 5.1. The three (orthogonally invariant) operators
+L := ⟨x, ∂y⟩ − 1
+2∆z
+R := ⟨y, ∂x⟩ + 1
+2|z|2
+E := Ey − Ex + Ez + n
+2
+give rise to yet another copy of the Lie algebra sl(2). This Lie algebra com-
+mutes with the Lie algebra sl(2) ∼= Alg(Ds, Xs).
+This thus means that we have now obtained a specific realisation for the Lie
+algebra so(4) ∼= Alg(Ds, Xs) ⊕ Alg(L, R) ∼= sl(2) ⊕ sl(2) which appears as a
+subalgebra of sp(6). This algebra will play the role of s from Section 4. Let
+us therefore consider the lowest weight vectors in so(4):
+Y1 = Ds = ⟨z, ∂y⟩ − ⟨∂z, ∂x⟩
+and
+Y2 = L = ⟨x, ∂y⟩ − 1
+2∆z .
+We will focus on the solutions of both lowest weight vectors, i.e. ker(Ds, L).
+Note that the operators in sp(6) do not necessarily act as endomorphisms
+on this space, but the transvector framework allows us to ‘replace’ these
+operators by (related) transvector algebra generators which do act as endo-
+morphisms. We start with proving the reductiveness of the algebra so(4) in
+sp(6).
+Lemma 5.2. The Lie algebra so(4) is reductive in sp(6).
+Proof. We need to show that sp(6) decomposes as so(4) + t, where the sub-
+space t carries an action of so(4). For that purpose we introduce the following
+15 (linearly independent) differential operators:
+∆x
+⟨z, ∂x⟩
+⟨y, ∂x⟩ − |z|2
+⟨y, z⟩
+|y|2
+⟨∂x, ∂y⟩
+⟨z, ∂y⟩ + ⟨∂z, ∂x⟩
+Ex − Ey + 2Ez + m
+⟨x, z⟩ − ⟨y, ∂z⟩
+⟨x, y⟩
+∆y
+⟨∂y, ∂z⟩
+⟨x, ∂y⟩ + ∆z
+⟨x, ∂z⟩
+|x|2
+It is now a straightforward computation to check that for each of these op-
+erators the commutator with one of the operators in so(4) is again a linear
+combination of the operators above.
+□
+In order to construct the generators for the algebra Z(g, s) with g = sp(6)
+and s = so(4), we need the following:
+Definition 5.3. The extremal projector for the Lie algebra sl(2) = Alg(X, Y, H)
+is the idempotent operator π given by the (formal) expression
+π := 1 +
+∞
+�
+j=1
+(−1)j
+j!
+Γ(H + 2)
+Γ(H + 2 + j)Y jXj .
+(5.1)
+This operator satisfies Xπ = πY = 0 and π2 = π.
+Note that this operator is defined on the extension U′(sl(2)) of the universal
+enveloping algebra defined earlier, so that formal series containing the oper-
+ator H in the denominator are well-defined (in practice it will always reduce
+to a finite summation).
+
+Branching symplectic monogenics using a M–Z algebra
+9
+Lemma 5.4. The extremal projector πso(4) is given by the product of the
+extremal projectors for the Lie algebras sl(2), i.e. πso(4) = πDsπL = πLπDs
+(the operator appearing as an index here refers to the realisation for sl(2)
+that was used).
+Proof. This is due to the fact that the two copies of sl(2) commute.
+□
+The operator πso(4) is thus explicitly given by
+
+1 +
+∞
+�
+j=1
+(−1)j
+j!
+Γ(E + 2)
+Γ(E + 2 + j)Xj
+sDj
+s
+
+
+
+1 +
+∞
+�
+j=1
+(−1)j
+j!
+Γ(E + 2)
+Γ(E + 2 + j)RjLj
+
+
+and satisfies Dsπso(4) = Lπso(4) = 0 = πso(4)Xs = πso(4)R. This means that
+we now have a natural object that can be used to project polynomials on the
+intersection of the kernel of the operators Ds and L.
+The 15 operators in t ⊂ sp(6) as such do not preserve this kernel space
+(as these operators do not necessarily commute with Ds and L), but their
+projections will belong to End(ker(Ds, L)). In what follows we will use the
+notation Qa,b, where a ∈ {±2, 0} and b ∈ {±4, ±2, 0}, to denote the operators
+in t (see Lemma 5.2, and the scheme below). For each operator Qa,b we then
+also define an associated operator Pa,b := πso(4)Qa,b. For instance P4,−2 =
+πso(4)|y|2.
+The P-operators will then be used to define the generators for our
+transvector algebra. The diagram below should then be seen as the analogue
+of the 15 operators Qa,b given above, grouped into a 5 × 3 rectangle, where
+each operator α ∈ t carries a label. The meaning of the labels (a, b) comes
+from the observation that t ∼= V4 ⊗ V2 as a representation for sl(2) ⊕ sl(2),
+with Vn the standard notation for the irreducible representation of dimension
+(n+1). Given an operator α ∈ t, the numbers a and b can thus be retrieved as
+eigenvalues for the commutator action of the Cartan elements in so(4). Note
+that the projection operator so(4) commutes with these Cartan elements (i.e.
+the operators Qa,b and Pa,b indeed carry the same labels).
+−2
+0
+2
+−4
+− 2
+0
+2
+4
+Despite the fact that Z(sp(6), so(4)) is not a Lie algebra, we have organised
+these operators in such a way that the notions of ‘positive’ and ‘negative’
+roots can be used. To be more precise: black dots (resp. grey dots) refer
+to negative (resp. positive) operators, and the white dot plays the role of
+a ‘Cartan element’ (this analogy will come in handy below). The 7 black
+
+10
+David Eelbode and Guner Muarem
+dots (resp. 7 grey dots) will be referred to as operators in ρ− (resp. in ρ+).
+Together with the operator P0,0 we then get the set
+GZ = {Pa,b : a ∈ {±2, 0}, b ∈ {±4, ±2, 0}},
+containing all the generators for the transvector algebra Z(sp(6), so(4)).
+Due to a general result by Zhelobenko, these generators then satisfy
+quadratic relations (i.e. different from the classical Lie brackets). In the next
+theorem, we will relate the spaces Ha,b,c(R3m, C) introduced in Definition
+3.3 to the space of polynomial solutions for the symplectic Dirac operator
+Ds, the lowering operator L and the negative ‘roots’ ρ− which we have just
+introduced (i.e. the operators Pa,b corresponding to black dots).
+Theorem 5.5. The solutions for the operators Ds and L and the negative
+roots ρ− ⊂ GZ which are homogeneous of degree (a, b, c) in the variables
+(z, x, y) are precisely given by the simplicial harmonics Ha,b,c(R3m, C). In
+other words, we have:
+Pa,b,c(R3m, C) ∩ ker(Ds, L, ρ−) = Ha,b,c(R3m, C).
+Proof. The idea behind this proof is a recursive argument, where the or-
+dering on the black dots will be from left to right and from bottom to
+top in the rectangular scheme above (in terms of labels this means that
+(2, −4) > (0, −4) > (2, −2), as an example). The reason for doing so is the
+following: the commutators [L, Qa,b] and [Ds, Qa,b] give an operator situated
+below or to the left of the operator Qa,b we started from. Up to a con-
+stant, these operators are equal to Qa+2,b and Qa,b−2 respectively (or trivial
+whenever the parameters a and b are not in the correct range). This means
+that combinations of the form LQa,b and DsQab act trivially on functions
+H(z; x, y) in the kernel of L and Ds, provided we know that also Qa+2,b and
+Qa,b−2 act trivially. Given the fact that each operator Pa,b ∈ ρ− is of the
+form
+Pa,b =
+�
+1 + O1L
+��
+1 + O2Ds
+�
+Qa,b ,
+where Oj is a short-hand notation for the correction terms coming from
+the extremal projection operator (which, unless this operator reduces to the
+identity operator, always contains either an operator L or Ds at the right).
+The upshot of our recursive scheme is that once we know that Qa+2,b and
+Qa,b−2 act trivially, this immediately tells us that Pa,bH = 0 ⇒ Qa,bH = 0.
+Because P2,−4H = 0 and P2,−4 = Q−2,4 = ∆y, we can immediately conclude
+that the following operators will then act trivially:
+∆y
+⟨∂x, ∂y⟩
+∆x
+⟨∂y, ∂z⟩
+⟨z, ∂y⟩ + ⟨∂x, ∂z⟩
+⟨z, ∂x⟩
+⟨x, ∂y⟩ + ∆z .
+In order to be simplicial harmonic, H(z; x, y) should belong to the kernel of
+9 operators in sp(6) (see Definition 3.3), but it is straightforward to see that
+one can reproduce these operators as commutators of the 7 operators on the
+previous line. For example: ∆x(⟨x, ∂y⟩ + ∆z)H = 0 leads to ∆zH = 0, since
+⟨∂x, ∂y⟩H = 0 (and so on).
+□
+
+Branching symplectic monogenics using a M–Z algebra
+11
+6. Application: branching symplectic monogenics
+We will now use the operators Pa,b to explicitly describe the branching of the
+k-homogeneous symplectic monogenics S∞
+k . By this we mean that it will give
+us a systematic way to define the ‘embedding factors’ realising the isomorphic
+copy of those spaces in S∞
+k . To do so, we will make an analogy again: one can
+consider the asssociative algebra U(Z), the ‘universal enveloping algebra’ of
+Z(sp(6), so(4)). The meaning should be clear here: it is a tensor algebra � V
+(with V the span of GZ-generators as an underlying vector space) modulo
+the ideal spanned ‘by the quadratic relations’ in the transvector algebra. We
+will refer to elements in this algebra as ‘words’ in ‘an alphabet’ that can be
+ordered. This statement, which should thus be seen as an analogue of the
+Poincar´e–Birkhoff–Witt theorem (PBW theorem), requires a proof but we
+will not do this in the present paper. As a matter of fact, the general case
+k ∈ Z+ will be treated in an upcoming (longer) paper, in the present article
+we will focus on the case k = 1 as a guiding example.
+The main idea is the following: imposing the lexicographic ordering on
+the labels (a, b) will dictate the position of our letters in the alphabet (from
+left to right), with e.g. (4, 0) > (4, −2) > (2, 2). Letting such a word acting as
+an operator on simplicial harmonics Ha,b,c(z; x, y), it should be clear (in view
+of the previous theorem) that only the ‘letters’ corresponding to grey dots
+in the scheme will play a role (the white dot acts as a constant, whereas the
+black dots act trivially). Considering the fact that the total degree of ‘a word’
+in x and y should not exceed k = 1, we can only use the operators Pa,b from
+the third and fourth column in our example. Note that once the operator
+Pab has been chosen (i.e.
+the ‘word’ in front of the simplicial harmonics),
+the degree (a, b, c) of these polynomials Ha,b,c(z; x, y) is automatically fixed
+too: the total degree in z and (x, y) is then equal to k and 1 respectively. So,
+when the ‘word’ is homogeneous of degree one in (x, y) we get contributions
+of the form P0,0Ha,1,0 and P2,0Ha,1,0. Whereas when the chosen ‘word’ is
+homogeneous of degree zero we get P−2,2Ha,0,0, P0,2Ha,0,0 and P2,2Ha,0,0.
+Finally, we note that we can still act with the raising operator R ∈ sl(2) on
+each of the polynomials from above (i.e. a suitable projection operator acting
+on a suitable space of simplicial harmonics) to arrive at a direct sum of
+Verma modules which can be embedded into S∞
+1 . This is based on the trivial
+albeit crucial observation that [R, Ds] = 0, so that acting with R preserves
+symplectic monogenic solutions. This means that we have now resolved the
+branching problem for k = 1 in a completely different way. Resulting in the
+decomposition
+S∞
+1
+�
+sp(2m)
+so(m)
+∼=
+�
+a≥1
+∞
+�
+ℓ=0
+Rℓ(Ha,1 ⊕ P2,0Ha,1)
+⊕
+�
+a≥0
+∞
+�
+ℓ=0
+Rℓ(P−2,2Ha ⊕ P−2,0Ha ⊕ P−2,−2Ha).
+
+12
+David Eelbode and Guner Muarem
+Summarising the idea behind this decomposition, we thus claim that S∞
+k can
+be decomposed under the joint action of
+so(m) × sl(2) × Z(sp(6), so(4)),
+whereby the final decomposition will contain summands of the form
+Rp �
+U(ρ+)Ha,b,c
+�
+for suitable ‘words’ in the algebra U(ρ+) and suitable spaces of simplicial
+harmonics.
+Acknowledgments
+The author G.M. was supported by the FWO-EoS project G0H4518N.
+References
+[1] F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis. Research Notes in
+Mathematics 76, Pitman, London, 1982.
+[2] H. De Bie, M. Hol´ıkov´a, P. Somberg, Basic aspects of symplectic Clifford analysis
+for the symplectic Dirac operator. Advances in Applied Clifford Algebras 27(2)
+(2017), 1103–1132.
+[3] D. Eelbode, G. Muarem, The Orthogonal Branching Problem for Symplectic
+Monogenics. Advances in Applied Clifford Algebras 33(3) (2022).
+[4] J. Gilbert, M. Murray, Clifford Algebras and Dirac Operators in Harmonic Anal-
+ysis. Cambridge University Press, 1991
+[5] K. Habermann, L. Habermann, Introduction to Symplectic Dirac Operators. In
+Lecture Notes in Mathematics, Springer Berlin Heidelberg, 2006.
+[6] R. Howe, Remarks on classical invariant theory. Transactions of the American
+Mathematical Society 33(2) (1989), 539—570
+[7] P. Robinson, J. Rawnsley, The Metaplectic Representation, Mpc Structures and
+Geometric Quantization. Memoirs of the A.M.S. vol. 81, no. 410, 1989.
+[8] P. Van Lancker, F. Sommen, D. Constales, Models for irreducible representations
+of Spin(m). Advances in Applied Clifford Algebras 11 (2001), 271–289.
+[9] D. Zhelobenko, Extremal projectors and generalised Mickelsson algebras over
+reductive Lie algebras. In Mathematics of the USSR 33(1) (1989), 85—100.
+David Eelbode
+Department of Mathematics
+University of Antwerp
+Middelheimlaan 1
+2020 Antwerp, Belgium
+e-mail: [email protected]
+Guner Muarem
+Department of Mathematics
+University of Antwerp
+Middelheimlaan 1
+2020 Antwerp, Belgium
+e-mail: [email protected]
+

2tE4T4oBgHgl3EQfawwa/content/tmp_files/load_file.txt

@@ -0,0 +1,339 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf,len=338
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='05066v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='RT]  12 Jan 2023  Branching symplectic monogenics using  a Mickelsson–Zhelobenko algebra  David Eelbode and Guner Muarem  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In this paper we consider (polynomial) solution spaces for  the symplectic Dirac operator (with a focus on 1-homogeneous solu-  tions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This space forms an infinite-dimensional representation space for  the symplectic Lie algebra sp(2m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Because so(m) ⊂ sp(2m), this leads  to a branching problem which generalises the classical Fischer decom-  position in harmonic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Due to the infinite nature of the solution  spaces for the symplectic Dirac operators, this is a non-trivial question:  both the summands appearing in the decomposition and their explicit  embedding factors will be determined in terms of a suitable Mickelsson-  Zhelobenko algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Primary 15A66, 17B10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Secondary 00A00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Branching, Symplectic Dirac operator, Mickelsson–Zhelobenko  algebra, simplicial harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Introduction  The Dirac operator is a first-order differential operator acting on spinor-  valued functions which factorises the Laplace operator ∆ on Rm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' It was  originally introduced by Dirac in a famous attempt to factorise the wave op-  erator, hence obtaining a relativistically invariant version of the Schr¨odinger  equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Since then, this operator has played a crucial role in mathemati-  cal domains such as representation theory and Clifford analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The latter  is a multidimensional function theory which is often described as a refine-  ment of harmonic analysis, and a generalisation of complex analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' It is  centred around a generalisation of the operator introduced by Dirac (his  operator /∂ is defined in 4 dimensions), and can be seen as a contraction  between the generators ek for a Clifford algebra (acting as endomorphisms  on so-called spinors) and corresponding partial derivatives ∂xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' To be more  precise, introducing the Clifford algebra by means of the defining relations  2  David Eelbode and Guner Muarem  {ea, eb} = eaeb + ebea = −2δab (with 1 ≤ a, b ≤ m) the Dirac operator is  given by  ∂x =  �  e1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  em  �  Idm  \uf8eb  \uf8ec  \uf8ed  x1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  xm  \uf8f6  \uf8f7  \uf8f8 =  m  �  j=1  ej∂xj ,  whereby the (m × m)−identity matrix Idm has been added to explain what  is meant by the ‘contraction’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Null-solutions for ∂x are called monogenics,  and can be seen as generalisations of holomorphic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' One often starts  with the study of k-homogeneous polynomial solutions for the Dirac operator,  which belong to the space Mk(Rm, S), where S stands for the aforementioned  spinor space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  An obvious generalisation of the operator ∂x can be obtained by using  another matrix than Idm when contracting algebraic generators with partial  derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' An important example is the symplectic Dirac operator, which  is introduced on a symplectic space rather than an orthogonal space (see for  example the work of Habermann [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This operator, denoted by Ds, is de-  fined as a contraction between generators for a symplectic Clifford algebra  and partial derivatives, using a skew-symmetric matrix Ω0 (rather than Idm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The symplectic Clifford algebra generators satisfy the Heisenberg relations  [∂zj, zk] = δjk (the symplectic analogue of the Clifford relations for the gener-  ators ek from above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Note that the symbols zj stand for real variables here,  they are chosen because the sets of (real) variables xj and yj will also appear  in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In sharp contrast to the orthogonal case, the symplectic Clifford  algebra is no longer finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This trend continues, in the sense that  the associated symplectic spinor space S∞  0 also becomes infinite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  In this paper, we study infinite-dimensional spaces defined in terms of  solutions for the symplectic Dirac operator (generalised monogenics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' These  spaces can be defined algebraically  S∞  k = Ms  k(R2m, S∞  0 ) := Pk(R2m, C) ⊠ S∞  0  (k ∈ N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Here ⊠ denotes the Cartan product of the sp(2m)-representations Pk(R2m, C),  the kth-symmetric power of the fundamental vector representation (modelled  by polynomials), and the symplectic spinor space S∞  0 (also referred to as the  Segal-Shale-Weil representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' These spaces contain k-homogeneous S∞  0 -  valued solutions for the symplectic Dirac operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The behaviour of these  spaces as representations for sp(2m) is known (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' [1] and the references  therein), but in this paper we will look at these spaces as orthogonal repre-  sentation spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This is motivated by the fact that so(m) ⊂ sp(2m), which  means that we are dealing with a branching problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  In general, a branching problem can be described as follows: given a rep-  resentation ρ of a Lie algebra g and a subalgebra h, we would like to under-  stand how the representation ρ behaves as a h-representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This restricted  representation ρ|h will no longer be irreducible, but will decompose into h-  irreducible representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' A branching rule then describes the irreducible  Branching symplectic monogenics using a M–Z algebra  3  pieces which will occur, together with their multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' For the symplec-  tic spinors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' for the space S∞  0 ), this gives the Fischer decomposition in  harmonic analysis, which means that the branching problem for S∞  k leads to  generalisations thereof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' To describe the branching of the infinite-dimensional  symplectic representation space S∞  k  under the inclusion so(m) ⊂ sp(2m),  we will make use of a quadratic algebra which is known as a Mickelson-  Zhelobenko algebra (see [9] for the general construction and properties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The symplectic Dirac operator and monogenics  We will work with the symplectic space R2m and coordinates (x, y) equipped  with the canonical symplectic form ω0 = �m  j=1 dxj ∧ dyj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The matrix repre-  sentation of the symplectic form is given by  Ω0 =  �  0  Idm  −Idm  0  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The group consisting of all invertible linear transformations preserving this  non-degenerate skew-symmetric bilinear form is called the symplectic group  and is formally defined as follows:  Sp(2m, R) = {M ∈ GL(2m, R) | M T Ω0M = Ω0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  This is a non-compact group of dimension 2m2+m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Its (real) Lie algebra will  be denoted by sp(2m, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In the orthogonal case, the spin group determined  by the sequence  1 → Z2 → Spin(m) → SO(m) → 1  plays a crucial role concerning the invariance of the Dirac operator ∂x and  the definition of the spinors S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In the symplectic case, this role is played by  the metaplectic group Mp(2m, R) fixed by the exact sequence  1 → Z2 → Mp(2m, R) → Sp(2m, R) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Despite the analogies, there are some fundamental differences:  (i) First of all, the group SO(m) is compact, whereas Sp(2m, R) is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This  has important consequences for the representation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' As a matter  of fact, the metaplectic group is not a matrix group and does not admit  (faithful) finite-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  (ii) The orthogonal spinors S can be realised as a maximal left ideal in the  Clifford algebra, but this is not the case for the symplectic spinors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The  latter are often modelled as smooth vectors in the infinite-dimensional  Segal-Shale-Weil representation (see [7] and the references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' One  can also identify the symplectic spinor space S∞  0 with the space P(Rm, C)  of polynomials in the variables (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' , zm) ∈ Rm, which is the approach  we will use in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Let (V, ω) be a symplectic vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The symplectic  Clifford algebra Cls(V, ω) is defined as the quotient algebra of the tensor  algebra T (V ) of V by the two-sided ideal Iω := {v ⊗ u − u ⊗ v + ω(v, u) :  4  David Eelbode and Guner Muarem  u, v ∈ V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In other words Cls(V, ω) := T (V )/Iω is the algebra generated by  V in terms of the relation [v, u] = −ω(v, u), where we have omitted the tensor  product symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Denote by ⟨u, v⟩ := �m  k=1 ukvk the canonical inner product  on Rm (where we allow partial derivatives to appear as coefficients, see the  operators below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' We then define the following operators acting on polyno-  mial functions in P(R3m, C):  (i) The symplectic Dirac operator Ds = ⟨z, ∂y⟩ − ⟨∂x, ∂z⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  (ii) The adjoint operator Xs = ⟨y, ∂z⟩+⟨x, z⟩ with respect to the symplectic  Fischer product (see Section 5 of [2] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  (iii) The Euler operator E = �m  j=1(xj∂xj + yj∂yj) = Ex + Ey measuring the  degree of homogeneity in the base variables (x, y) ∈ R2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Note that some authors use the notation ⟨∇x, ∇y⟩ for an expression such as  �  k ∂xk∂yk, but we will use the Dirac operator symbol here instead of the  nabla operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The three operators X =  √  2Ds, Y =  √  2Xs and their commu-  tator H = [X, Y ] = −2(Ex + Ey + m) give rise to a copy of the Lie algebra  sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  One now easily sees that the symplectic Dirac operator is nothing more than  the contraction between the Weyl algebra generators (zk, ∂zk) with the vector  fields (∂xk, ∂yk) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' , m using the canonical symplectic form Ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The space of k-homogeneous symplectic monogenics is de-  fined by S∞  k := ker(Ds)∩  �  Pk(R2m, C) ⊗ P(Rm, C)  �  , where the space P(Rm, C)  in the vector variable z ∈ Rm plays the role of the symplectic spinor space  S∞  0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Note that as an sp(2m, R)-module, S∞  k is reducible and decomposes into two  irreducible parts: S∞  k = S∞  k,+ ⊕ S∞  k,− with highest weights  S∞  k,+ ←→  �  k − 1  2, −1  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' , −1  2  �  and  S∞  k,+ ←→  �  k − 1  2, −1  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' , −3  2  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  These weight entries are fixed by the Cartan algebra h = Alg(Xjj : 1 ≤ j ≤  m), where the elements Xjj are defined in the lemma below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In this paper, we  will omit the parity signs and work with S∞  k as a notation which incorporates  both the positive and negative spinors (in our model, this will correspond to  even or odd in the variable z ∈ Rm, see below, so it is always easy to ‘decom-  pose’ into irreducible components when necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The three operators from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='3 can be proven to be invariant under the  action of the symplectic Lie algebra, in the sense that they commute with  the following generators (see also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='3 in [3]):  Branching symplectic monogenics using a M–Z algebra  5  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The symplectic Lie algebra sp(2m) has the following realisation  on the space of symplectic spinor-valued polynomials P(R2m, C) ⊗ S∞  0 :  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  Xjk = xj∂xk − yk∂yj − (zk∂zj + 1  2δjk)  1 ≤ j, k ≤ m  Yjk = xj∂yk + xk∂yj − ∂zj∂zk  1 ≤ j < k ≤ m  Zjk = yj∂xk + yk∂xj + zjzk  1 ≤ j < k ≤ m  Yjj = xj∂yj − 1  2∂2  zj  1 ≤ j ≤ m  Zjj = yj∂xj + 1  2z2  j  1 ≤ j ≤ m  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='1)  The branching rule for S∞  0 , when considering it as a representation space for  the orthogonal Lie algebra so(m) ⊂ sp(2m), leads to the Fischer decomposi-  tion for C-valued polynomials in the variable z ∈ Rm (see below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Note that  so(m) is generated by the operators Xjk −Xkj for 1 ≤ j < k ≤ m, giving rise  to the well-known angular operators ubiquitous in quantum mechanics (often  denoted by Lab with 1 ≤ a < b ≤ m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In our previous paper [3], we therefore  tackled the next case k = 1 as this is a natural generalisation of said Fischer  decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The main problem with our branching rule (Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='6 in  [3]) is the fact that these so(m)-spaces appear with infinite multiplicities,  which are not always easy to keep track of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Therefore the main goal of this  paper is to show that one can organise these in an algebraic framework which  extends to other values for k too, using a certain quadratic algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Simplicial harmonics in three vector variables  In this section we describe a generalisation of harmonic polynomials, in three  vector variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This will be done in terms of a solution space for a ‘natural’  collection of so(m)-invariant differential operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The corresponding Howe  dual pair will be useful for the branching problem addressed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' For the  sake of completeness, we recall the following basic definition:  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' A function f(x) on Rm is called harmonic if ∆f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The  k-homogeneous harmonics are defined as Hk(Rm, C) := Pk(Rm, C) ∩ ker(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  These spaces define irreducible representations for so(m) with highest weight  (k, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' , 0) for all k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  It is well-known that the space of k-homogeneous polynomials Pk(Rm, C) is  reducible as an so(m)-module (see for example [4]) and decomposes into har-  monic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In fact, the decomposition of the full space of polynomials  is known as the aforementioned Fischer decomposition, given by  P(Rm, C) =  ∞  �  k=0  Pk(Rm, C) =  ∞  �  k=0  ∞  �  p=0  |z|2pHk(Rm, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  This can all be generalised to the case of several vector variables (sometimes  also called ‘a matrix variable’): for any highest weight for so(m) there is a  (polynomial) model in terms of simplicial harmonics (or monogenics for the  half-integer representations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' We refer to [8] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In this paper,  6  David Eelbode and Guner Muarem  we will consider these spaces for so(m)-weights characterised by three inte-  gers (a, b, c) where a ≥ b ≥ c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Also note that trailing zeros in the weight  notation will be omitted from now on, so for instance (k, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' , 0) will be writ-  ten as (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' First of all, we consider homogeneous polynomials Pa,b,c(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y)  in three vector variables (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y) ∈ R3m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Here we use the notation (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y)  to stress the difference between the variable z (the spinor variable, refer-  ring to an element in S∞  0 ) from the other two variables (x, y) ∈ R2m, which  are ‘ordinary’ variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The parameters (a, b, c) then refer to the degrees of  homogeneity in (z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' These polynomials carry the regular representation  of the orthogonal group (or the derived so(m)-action in terms of angular  momentum operators Lab from above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  We further introduce the Weyl algebra in three vector variables as the  algebra generated by the variables and their corresponding derivatives:  W(R3m, C) := Alg(xα, yβ, zγ, ∂xδ, ∂yε, ∂zζ) with α, β, γ, δ, ε, ζ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=', m} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Just like in the case of the classical Fischer decomposition, where the Lie  algebra sl(2) appears as a Howe dual partner, there is a Lie algebra appearing  here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' To be precise, it is the Lie algebra sp(6) = g−2⊕g0⊕g+2, with parabolic  subalgebra p := g−2 ⊕ g0 and Levi subalgebra g0 ∼= gl(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The subspaces g±2  contain six ‘pure’ operators each (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' only variables, acting as a multiplication  operator, or only derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' More specifically, the subspaces are spanned  by the following SO(m)-invariant operators:  g−2 := span(∆x, ∆y, ∆z, ⟨∂x, ∂y⟩, ⟨∂y, ∂z⟩, ⟨∂x, ∂z⟩)  g0 := span(⟨x, ∂y⟩, ⟨y, ∂x⟩, ⟨x, ∂z⟩, ⟨z, ∂x⟩, ⟨y, ∂z⟩, ⟨z, ∂y⟩, Ex, Ey, Ez)  g+2 := span(|x|2, |y|2, |z|2, ⟨x, y⟩, ⟨y, z⟩, ⟨x, z⟩)  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The space of Howe harmonics of degree (a, b, c) in the vari-  ables (z, x, y) is defined as H∗  a,b,c(R3m, C) := Pa,b,c(R3m, C) ∩ ker(g−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  In what follows the notation ker(A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' , An) stands for ker(A1)∩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='∩ker(An),  so ker(g−2) means that simplicial harmonics are annihilated by all (pure dif-  ferential) operators in sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' As a representation space for so(m), the spaces  H∗  a,b,c are not irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In order to obtain an irreducible (sub)space, we  have to impose extra conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The vector space of simplicial harmonics of degree (a, b, c)  in the variables (z, x, y) is defined by means of  Ha,b,c(R3m, C) := H∗  a,b,c(R3m, C) ∩ ker  �  ⟨z, ∂x⟩, ⟨z, ∂y⟩, ⟨x, ∂y⟩  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  As was shown in [8], this defines an irreducible representation space for so(m)  with highest weight (a, b, c), where the dominant weight condition a ≥ b ≥ c  must hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This now leads to the following generalisation of the result above  (the Fisher decompostion in three vector variables):  Branching symplectic monogenics using a M–Z algebra  7  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The space P(R3m, C) of complex-valued polynomials in three  vector variables (in Rm) has a multiplicity-free decomposition under the ac-  tion of sp(6) × SO(m) by means of:  P(R3m, C) ∼=  �  a≥b≥c  V∞  a,b,c ⊗ Ha,b,c(R3m, C),  where we used the dominant weight condition in the summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The notation  V∞  a,b,c hereby refers to a Verma module (see for example [6]) for sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The Mickelsson-Zhelobenko algebra (general setup)  We have now introduced 21 differential operators giving rise to a realisation  of the Lie algebra sp(6) inside the Weyl algebra (on 3 vector variables in  Rm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In this section we construct a related algebra, the so-called Mickelsson-  Zhelobenko algebra (also called transvector or step algebra) Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Let g be  a Lie algebra and let s ⊂ g be a reductive subalgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' We then have the  decomposition g = s ⊕ t, where t carries an s-action for the commutator (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  [s, t] ⊂ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' For s we then fix a triangular decomposition s = s− ⊕ h ⊕ s+,  where s± consists of the positive (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' negative roots) with respect to the  Cartan subalgebra h ⊂ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' We then also define a left ideal J ⊂ U(g) in the  universal enveloping algebra U(g) by means of U(g)s+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This allows us to  define a certain subalgebra of U(g) which is known as the normaliser:  Norm(J) := {u ∈ U(g) | Ju ⊂ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The crucial point is that J is a two-sided ideal of Norm(J), which allows us  two define the quotient algebra S(g, s) = Norm(J)/J which is known as the  Mickelsson algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  In a last step of the construction, we consider an extension of U(g) to a  suitable localisation U′(g) given by  U′(g) = U′(g) ⊗U(h) Frac(U(h)) ,  where Frac(U(h)) is the field of fractions in the (universal enveloping algebra  of the) Cartan algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The ideal J′ can be introduced for this extension  too (in a completely similar way) and the corresponding quotient algebra  Z(g, s) := Norm(J′)/J′ is the Mickelsson-Zhelobenko algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' These two al-  gebras are naturally identified, since one has that  Z(g, s) = S(g, s) ⊗U(h) Frac(U(h)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Note that this algebra is sometimes referred to as a ‘transvector algebra’,  which is what we will often use in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The Mickelsson-Zhelobenko algebra Z(sp(6), so(4))  We will now define a specific example of the construction from above, which  will help us to understand how the branching of S∞  k  works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' First of all, we  note the following:  8  David Eelbode and Guner Muarem  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The three (orthogonally invariant) operators  L := ⟨x, ∂y⟩ − 1  2∆z  R := ⟨y, ∂x⟩ + 1  2|z|2  E := Ey − Ex + Ez + n  2  give rise to yet another copy of the Lie algebra sl(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This Lie algebra com-  mutes with the Lie algebra sl(2) ∼= Alg(Ds, Xs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  This thus means that we have now obtained a specific realisation for the Lie  algebra so(4) ∼= Alg(Ds, Xs) ⊕ Alg(L, R) ∼= sl(2) ⊕ sl(2) which appears as a  subalgebra of sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This algebra will play the role of s from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Let  us therefore consider the lowest weight vectors in so(4):  Y1 = Ds = ⟨z, ∂y⟩ − ⟨∂z, ∂x⟩  and  Y2 = L = ⟨x, ∂y⟩ − 1  2∆z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  We will focus on the solutions of both lowest weight vectors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' ker(Ds, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Note that the operators in sp(6) do not necessarily act as endomorphisms  on this space, but the transvector framework allows us to ‘replace’ these  operators by (related) transvector algebra generators which do act as endo-  morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' We start with proving the reductiveness of the algebra so(4) in  sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The Lie algebra so(4) is reductive in sp(6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' We need to show that sp(6) decomposes as so(4) + t, where the sub-  space t carries an action of so(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' For that purpose we introduce the following  15 (linearly independent) differential operators:  ∆x  ⟨z, ∂x⟩  ⟨y, ∂x⟩ − |z|2  ⟨y, z⟩  |y|2  ⟨∂x, ∂y⟩  ⟨z, ∂y⟩ + ⟨∂z, ∂x⟩  Ex − Ey + 2Ez + m  ⟨x, z⟩ − ⟨y, ∂z⟩  ⟨x, y⟩  ∆y  ⟨∂y, ∂z⟩  ⟨x, ∂y⟩ + ∆z  ⟨x, ∂z⟩  |x|2  It is now a straightforward computation to check that for each of these op-  erators the commutator with one of the operators in so(4) is again a linear  combination of the operators above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  □  In order to construct the generators for the algebra Z(g, s) with g = sp(6)  and s = so(4), we need the following:  Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The extremal projector for the Lie algebra sl(2) = Alg(X, Y, H)  is the idempotent operator π given by the (formal) expression  π := 1 +  ∞  �  j=1  (−1)j  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Γ(H + 2)  Γ(H + 2 + j)Y jXj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='1)  This operator satisfies Xπ = πY = 0 and π2 = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Note that this operator is defined on the extension U′(sl(2)) of the universal  enveloping algebra defined earlier, so that formal series containing the oper-  ator H in the denominator are well-defined (in practice it will always reduce  to a finite summation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Branching symplectic monogenics using a M–Z algebra  9  Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The extremal projector πso(4) is given by the product of the  extremal projectors for the Lie algebras sl(2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' πso(4) = πDsπL = πLπDs  (the operator appearing as an index here refers to the realisation for sl(2)  that was used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This is due to the fact that the two copies of sl(2) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  □  The operator πso(4) is thus explicitly given by  \uf8eb  \uf8ed1 +  ∞  �  j=1  (−1)j  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Γ(E + 2)  Γ(E + 2 + j)Xj  sDj  s  \uf8f6  \uf8f8  \uf8eb  \uf8ed1 +  ∞  �  j=1  (−1)j  j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Γ(E + 2)  Γ(E + 2 + j)RjLj  \uf8f6  \uf8f8  and satisfies Dsπso(4) = Lπso(4) = 0 = πso(4)Xs = πso(4)R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This means that  we now have a natural object that can be used to project polynomials on the  intersection of the kernel of the operators Ds and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The 15 operators in t ⊂ sp(6) as such do not preserve this kernel space  (as these operators do not necessarily commute with Ds and L), but their  projections will belong to End(ker(Ds, L)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In what follows we will use the  notation Qa,b, where a ∈ {±2, 0} and b ∈ {±4, ±2, 0}, to denote the operators  in t (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='2, and the scheme below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' For each operator Qa,b we then  also define an associated operator Pa,b := πso(4)Qa,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' For instance P4,−2 =  πso(4)|y|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The P-operators will then be used to define the generators for our  transvector algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The diagram below should then be seen as the analogue  of the 15 operators Qa,b given above, grouped into a 5 × 3 rectangle, where  each operator α ∈ t carries a label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The meaning of the labels (a, b) comes  from the observation that t ∼= V4 ⊗ V2 as a representation for sl(2) ⊕ sl(2),  with Vn the standard notation for the irreducible representation of dimension  (n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Given an operator α ∈ t, the numbers a and b can thus be retrieved as  eigenvalues for the commutator action of the Cartan elements in so(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Note  that the projection operator so(4) commutes with these Cartan elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  the operators Qa,b and Pa,b indeed carry the same labels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  −2  0  2  −4  − 2  0  2  4  Despite the fact that Z(sp(6), so(4)) is not a Lie algebra, we have organised  these operators in such a way that the notions of ‘positive’ and ‘negative’  roots can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' To be more precise: black dots (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' grey dots) refer  to negative (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' positive) operators, and the white dot plays the role of  a ‘Cartan element’ (this analogy will come in handy below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The 7 black  10  David Eelbode and Guner Muarem  dots (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' 7 grey dots) will be referred to as operators in ρ− (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' in ρ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Together with the operator P0,0 we then get the set  GZ = {Pa,b : a ∈ {±2, 0}, b ∈ {±4, ±2, 0}},  containing all the generators for the transvector algebra Z(sp(6), so(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Due to a general result by Zhelobenko, these generators then satisfy  quadratic relations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' different from the classical Lie brackets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In the next  theorem, we will relate the spaces Ha,b,c(R3m, C) introduced in Definition  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='3 to the space of polynomial solutions for the symplectic Dirac operator  Ds, the lowering operator L and the negative ‘roots’ ρ− which we have just  introduced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' the operators Pa,b corresponding to black dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The solutions for the operators Ds and L and the negative  roots ρ− ⊂ GZ which are homogeneous of degree (a, b, c) in the variables  (z, x, y) are precisely given by the simplicial harmonics Ha,b,c(R3m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' In  other words, we have:  Pa,b,c(R3m, C) ∩ ker(Ds, L, ρ−) = Ha,b,c(R3m, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The idea behind this proof is a recursive argument, where the or-  dering on the black dots will be from left to right and from bottom to  top in the rectangular scheme above (in terms of labels this means that  (2, −4) > (0, −4) > (2, −2), as an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The reason for doing so is the  following: the commutators [L, Qa,b] and [Ds, Qa,b] give an operator situated  below or to the left of the operator Qa,b we started from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Up to a con-  stant, these operators are equal to Qa+2,b and Qa,b−2 respectively (or trivial  whenever the parameters a and b are not in the correct range).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This means  that combinations of the form LQa,b and DsQab act trivially on functions  H(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y) in the kernel of L and Ds, provided we know that also Qa+2,b and  Qa,b−2 act trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Given the fact that each operator Pa,b ∈ ρ− is of the  form  Pa,b =  �  1 + O1L  ��  1 + O2Ds  �  Qa,b ,  where Oj is a short-hand notation for the correction terms coming from  the extremal projection operator (which, unless this operator reduces to the  identity operator, always contains either an operator L or Ds at the right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The upshot of our recursive scheme is that once we know that Qa+2,b and  Qa,b−2 act trivially, this immediately tells us that Pa,bH = 0 ⇒ Qa,bH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Because P2,−4H = 0 and P2,−4 = Q−2,4 = ∆y, we can immediately conclude  that the following operators will then act trivially:  ∆y  ⟨∂x, ∂y⟩  ∆x  ⟨∂y, ∂z⟩  ⟨z, ∂y⟩ + ⟨∂x, ∂z⟩  ⟨z, ∂x⟩  ⟨x, ∂y⟩ + ∆z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  In order to be simplicial harmonic, H(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y) should belong to the kernel of  9 operators in sp(6) (see Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='3), but it is straightforward to see that  one can reproduce these operators as commutators of the 7 operators on the  previous line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' For example: ∆x(⟨x, ∂y⟩ + ∆z)H = 0 leads to ∆zH = 0, since  ⟨∂x, ∂y⟩H = 0 (and so on).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  □  Branching symplectic monogenics using a M–Z algebra  11  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Application: branching symplectic monogenics  We will now use the operators Pa,b to explicitly describe the branching of the  k-homogeneous symplectic monogenics S∞  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' By this we mean that it will give  us a systematic way to define the ‘embedding factors’ realising the isomorphic  copy of those spaces in S∞  k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' To do so, we will make an analogy again: one can  consider the asssociative algebra U(Z), the ‘universal enveloping algebra’ of  Z(sp(6), so(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' The meaning should be clear here: it is a tensor algebra � V  (with V the span of GZ-generators as an underlying vector space) modulo  the ideal spanned ‘by the quadratic relations’ in the transvector algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' We  will refer to elements in this algebra as ‘words’ in ‘an alphabet’ that can be  ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This statement, which should thus be seen as an analogue of the  Poincar´e–Birkhoff–Witt theorem (PBW theorem), requires a proof but we  will not do this in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' As a matter of fact, the general case  k ∈ Z+ will be treated in an upcoming (longer) paper, in the present article  we will focus on the case k = 1 as a guiding example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  The main idea is the following: imposing the lexicographic ordering on  the labels (a, b) will dictate the position of our letters in the alphabet (from  left to right), with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' (4, 0) > (4, −2) > (2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Letting such a word acting as  an operator on simplicial harmonics Ha,b,c(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y), it should be clear (in view  of the previous theorem) that only the ‘letters’ corresponding to grey dots  in the scheme will play a role (the white dot acts as a constant, whereas the  black dots act trivially).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Considering the fact that the total degree of ‘a word’  in x and y should not exceed k = 1, we can only use the operators Pa,b from  the third and fourth column in our example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Note that once the operator  Pab has been chosen (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  the ‘word’ in front of the simplicial harmonics),  the degree (a, b, c) of these polynomials Ha,b,c(z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' x, y) is automatically fixed  too: the total degree in z and (x, y) is then equal to k and 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' So,  when the ‘word’ is homogeneous of degree one in (x, y) we get contributions  of the form P0,0Ha,1,0 and P2,0Ha,1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Whereas when the chosen ‘word’ is  homogeneous of degree zero we get P−2,2Ha,0,0, P0,2Ha,0,0 and P2,2Ha,0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Finally, we note that we can still act with the raising operator R ∈ sl(2) on  each of the polynomials from above (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' a suitable projection operator acting  on a suitable space of simplicial harmonics) to arrive at a direct sum of  Verma modules which can be embedded into S∞  1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This is based on the trivial  albeit crucial observation that [R, Ds] = 0, so that acting with R preserves  symplectic monogenic solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' This means that we have now resolved the  branching problem for k = 1 in a completely different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Resulting in the  decomposition  S∞  1  \uf8e6\uf8e6\uf8e6�  sp(2m)  so(m)  ∼=  �  a≥1  ∞  �  ℓ=0  Rℓ(Ha,1 ⊕ P2,0Ha,1)  ⊕  �  a≥0  ∞  �  ℓ=0  Rℓ(P−2,2Ha ⊕ P−2,0Ha ⊕ P−2,−2Ha).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  12  David Eelbode and Guner Muarem  Summarising the idea behind this decomposition, we thus claim that S∞  k can  be decomposed under the joint action of  so(m) × sl(2) × Z(sp(6), so(4)),  whereby the final decomposition will contain summands of the form  Rp �  U(ρ+)Ha,b,c  �  for suitable ‘words’ in the algebra U(ρ+) and suitable spaces of simplicial  harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  Acknowledgments  The author G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' was supported by the FWO-EoS project G0H4518N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  References  [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Brackx, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Delanghe and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Sommen, Clifford Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Research Notes in  Mathematics 76, Pitman, London, 1982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='  [2] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' De Bie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content=' Hol´ıkov´a, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
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+page_content=' Zhelobenko, Extremal projectors and generalised Mickelsson algebras over  reductive Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
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+page_content='  David Eelbode  Department of Mathematics  University of Antwerp  Middelheimlaan 1  2020 Antwerp, Belgium  e-mail: david.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='eelbode@uantwerpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='be  Guner Muarem  Department of Mathematics  University of Antwerp  Middelheimlaan 1  2020 Antwerp, Belgium  e-mail: guner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='muarem@uantwerpen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}
+page_content='be' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/2tE4T4oBgHgl3EQfawwa/content/2301.05066v1.pdf'}

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+Adaptive Least-Squares Methods for Convection-Dominated
+Diffusion-Reaction Problems
+Zhiqiang Cai∗
+Binghe Chen†
+Jing Yang‡
+Abstract
+This paper studies adaptive least-squares finite element methods for convection-
+dominated diffusion-reaction problems. The least-squares methods are based on the
+first-order system of the primal and dual variables with various ways of imposing
+outflow boundary conditions. The coercivity of the homogeneous least-squares func-
+tionals are established, and the a priori error estimates of the least-squares methods
+are obtained in a norm that incorporates the streamline derivative. All methods have
+the same convergence rate provided that meshes in the layer regions are fine enough.
+To increase computational accuracy and reduce computational cost, adaptive least-
+squares methods are implemented and numerical results are presented for some test
+problems.
+ADAPTIVE FOSLS FOR THE CONVECTION-DOMINATED PROBLEMS
+1
+Introduction
+Due to the small diffusion coefficient, the solution of the convection-dominated diffusion-
+reaction problem develops the boundary or interior layers, i.e., narrow regions where
+derivatives of the solution change dramatically. It is well known that the conventional
+numerical methods do not work well on either stability or accuracy for such problems. For
+example, the standard Galerkin method with continuous linear elements exhibits large
+spurious oscillation in the boundary layer region.
+Over the decades, many successful
+numerical methods have been studied and may be roughly grouped into three categories:
+the mesh-fitted approach, the operator-fitted approach, and the stabilization approach.
+The mesh-fitted approach utilizes the a priori information of the solution including the
+location and the width of the layer to construct a layer-fitted mesh, e.g., the Shishkin
+mesh. The operator-fitted approach applies the layer-alike functions as the bases of the
+approximation space.
+The stabilization approach adds some stabilization term to the
+∗Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-
+2067, [email protected]. This work was supported in part by the National Science Foundation under
+grants DMS-1217081 and DMS-1522707.
+†Wells
+Fargo
+Corporate
+&
+Investment
+Banking,
+Charlotte,
+NC
+28202-4200,
+[email protected].
+‡School of Mathematical Science, Peking University, No.5 Yiheyuan Road Haidian District, Beijing,
+P.R.China 100871, [email protected].
+1
+arXiv:2301.11582v1  [math.NA]  27 Jan 2023
+
+2
+bilinear form. For example, the well-known streamline upwind Petrov-Galerkin (SUPG)
+method [21] adds the original equation tested by the convection term as the stabilization.
+For a comprehensive collection of the methods, see [23] and the references therein.
+Recently, least-squares methods have been intensively studied for fluid flow and elas-
+ticity problems (see, e.g., [5, 7, 8, 9, 12, 14, 15, 16]). The least-squares methods minimize
+certain norms of the residual of the first-order system over appropriate finite element
+spaces. The method always leads to a symmetric positive definite problem, and choices
+of finite element spaces for the primal and dual variables are not subject to the LBB
+condition. Moreover, one striking feature of the least-squares method is that the value of
+the least-squares functional at the current approximation provides an accurate estimates
+of the true error.
+The application of the least-squares methods to the convection-dominated diffusion-
+reaction problems is still in its infancy. Reported in [17] is a new least-squares formulation
+with inflow boundary conditions weakly imposed and outflow boundary conditions ultra-
+weakly imposed. This formulation works well on regions away from the boundary layer,
+even on coarse meshes. However, it does not resolve the boundary layer, which is the
+primary interest of the problem. This phenomena is also observed in the DG method [4],
+where the boundary conditions are weakly imposed. These works motivate us to treat
+outflow boundary conditions in different fashions. In particular, we study least-squares
+method for the convection-dominated diffusion-reaction problem with three different ways
+to handle the outflow boundary conditions. The a priori error estimates of finite element
+approximations based on these formulations are established.
+The solution of the convection-dominated diffusion-reaction problem usually consists
+of two parts: the solution of a transport problem (ϵ = 0) and the correction (i.e., the
+boundary layer). To compute the first part, it is sufficient to use a coarse mesh, while it
+requires a very fine mesh to resolve the boundary layer. Without the a priori information
+on locations of the layers, this observation motivates the use of adaptive mesh refinement
+algorithm, which has been vastly studied (see, e.g., [2, 3, 6, 13, 19, 24]). However, many a
+posteriori error estimators are not suitable for the convection-dominated diffusion-reaction
+problems, since they depend on the small diffusion parameter.
+To design a robust a
+posteriori error estimator is non-trivial. Nevertheless, for a least-squares formulation, the a
+posteriori error estimator is handy, which is simply the value of the least-squares functional
+at the current approximation. Since the least-squares functional has been computed when
+solving the algebraic equation, there is no additional cost. Besides, the reliability and
+the efficiency stem easily from the coercivity and the continuity of the bilinear form,
+respectively.
+In this paper, we present numerical results of adaptive mesh refinement
+algorithms using the least-squares estimator.
+The rest of this paper is organized as follows. In section 2, we present the convection-
+dominated diffusion-reaction problem and its first-order linear system. Based on the first-
+order system, three least-squares formulations are introduced and their coercivity are
+established in section 3. Section 4 is a computable counterpart of the previous section,
+which introduces the computable mesh dependent norms to replace the fractional norms
+in the least-squares functionals.
+The main objective of section 5 is to establish the a
+priori error estimates. The adaptive mesh refinement algorithm and the numerical tests
+
+3
+are exhibited in section 6 and section 7, respectively.
+1.1
+Notation
+We use the standard notation and definitions for the Sobolev spaces Hs(Ω)d and Hs(∂Ω)d
+for s ≥ 0. The standard associated inner products are denoted by (·, ·)s,Ω and (·, ·)s,∂Ω,
+and their respective norms are denoted by ∥·∥s,Ω and ∥·∥s,∂Ω. (We suppress the superscript
+d because the dependence on dimension will be clear by context. We also omit the subscript
+Ω from the inner product and norm designation when there is no risk of confusion.) For
+s = 0, Hs(Ω)d coincides with L2(Ω)d. In this case, the inner product and norm will be
+denoted by (·, ·) and ∥ · ∥, respectively. Finally, we define some spaces
+H1
+D(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD},
+H1
+D±(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD±},
+and
+H(div; Ω) = {v ∈ L2(Ω)2 : ∇ · v ∈ L2(Ω)},
+which is a Hilbert space under the norm
+∥v∥H(div; Ω) =
+�
+∥v∥2 + ∥∇ · v∥2� 1
+2 .
+2
+The convection-diffusion-reaction problem
+Let Ω be a bounded, open, connected subset in Rd (d = 2, 3) with a Lipschitz continu-
+ous boundary ∂Ω. Denote by n = (n1, · · · , nd)t the outward unit vector normal to the
+boundary. For a given vector-valued function β, denote by
+Γ+ = {x ∈ ∂Ω : β · n(x) > 0}
+and
+Γ− = {x ∈ ∂Ω : β · n(x) < 0}
+the outflow and inflow boundaries, respectively.
+Consider the following stationary convection-dominated diffusion-reaction problem:
+−ϵ ∆u + β · ∇u + c u = f
+in Ω,
+(2.1)
+where the diffusion coefficient ϵ is a given small constant, i.e., 0 < ϵ ≪ 1; and c and
+f are given scalar-valued functions. For simplicity, we consider homogeneous Dirichlet
+boundary condition:
+u|∂Ω = 0.
+(2.2)
+For the convection and reaction coefficients, we assume that:
+(1) β ∈ W 1
+∞(Ω)d and c ∈ L∞(Ω) with ∥c∥∞ ≤ γ;
+(2) there exists a positive constant α0 such that
+0 < α0 ≤ c − 1
+2∇ · β
+a.e. in Ω.
+(2.3)
+
+4
+Introducing the dual variable
+σ = −ϵ1/2∇u,
+(2.1) may be rewritten as the following first-order system:
+�
+σ + ϵ1/2∇u
+=
+0
+in Ω,
+ϵ1/2∇ · σ + β · ∇u + c u
+=
+f
+in Ω.
+(2.4)
+3
+Least-squares formulations
+In this section, we study three least-squares formulations based on the first-order system in
+(2.4) with the inflow boundary conditions imposed strongly. These formulations differ in
+how to handle the outflow boundary conditions. More specifically, the outflow boundary
+conditions are treated strongly for the first one and weakly for the other two through
+weighted boundary functionals.
+To this end, introduce the following least-squares functionals:
+G1(τ, v; f)
+=
+∥τ + ϵ1/2 ∇v∥2 + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2,
+(3.1)
+G2(τ, v; f)
+=
+G1(τ, v; f) + ∥ϵ−1/2 v∥2
+1/2,Γ+,
+(3.2)
+and
+G3(τ, v; f)
+=
+G1(τ, v; f) + ∥v∥2
+1/2,Γ+.
+(3.3)
+Since ϵ is very small, the outflow boundary conditions are enforced stronger in G2 than in
+G3. Let
+U1 = H(div; Ω) × H1
+0(Ω)
+and
+U2 = U3 = H(div; Ω) × H1
+Γ−(Ω).
+Then the least-squares formulations are to find (σ, u) ∈ Ui such that
+Gi(σ, u; f) =
+min
+(τ , v)∈ Ui
+Gi(τ, v; f)
+(3.4)
+for i = 1, 2, 3.
+For any (τ, v) ∈ Ui, define the following norms:
+M1(τ, v) = ∥τ∥2 + ∥v∥2 + ∥ϵ1/2 ∇v∥2,
+M2(τ, v) = M1(τ, v) + ∥ϵ−1/2 v∥2
+1/2,Γ+,
+and
+M3(τ, v) = M1(τ, v) + ∥v∥2
+1/2,Γ+.
+Below we show that the homogeneous least-squares functionals are coercive with respect
+to the corresponding norms. In particular, the coercivity of the functionals G1 and G2 are
+independent of the ϵ.
+Theorem 3.1 (Coercivity). For all (τ, v) ∈ Ui with i = 1, 2, 3, there exist positive
+constants Ci such that
+Mi(τ, v) ≤ Ci Gi(τ, v; 0),
+(3.5)
+where C1 and C2 are independent of the ϵ and C3 is proportional to ϵ−1/2.
+
+5
+Proof. We provide proofs for i = 2 and 3 in detail with an emphasis on how the weight in
+G2 leads to the coercivity constant independent of the ϵ. The case of i = 1 may be proved
+in a similar fashion as the case of i = 2.
+For all (τ, v) ∈ Ui with i = 1, 2, 3, the triangle inequality gives
+∥τ∥ ≤ ∥τ + ϵ1/2 ∇v∥ + ∥ϵ1/2 ∇v∥ ≤ G1/2
+1
+(τ, v; 0) + ∥ϵ1/2 ∇v∥.
+(3.6)
+Hence, to show the validity of (3.5), it suffices to prove that
+∥v∥2 + ∥ϵ1/2 ∇v∥2 ≤ Ci Gi(τ, v; 0)
+∀ (τ, v) ∈ Ui.
+(3.7)
+To this end, let
+I = −
+�
+ϵ1/2 ∇v, τ
+�
++
+�
+v, (c − 1
+2 ∇ · β) v
+�
++ 1
+2 ∥(β · n)1/2 v∥2
+0,Γ+.
+(3.8)
+It follows from the definition of the outflow boundary condition and the Cauchy-Schwarz
+inequality that
+∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ (ϵ1/2 ∇v, ϵ1/2∇v + τ) + I ≤ ∥ϵ1/2 ∇v∥ G1(τ, v; 0) + I,
+which implies
+∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C (G1(σ, u; 0) + I) .
+(3.9)
+To bound I, first note that integration by parts and the boundary conditions imply that
+(ϵ1/2 ∇v, τ)
+=
+(v, ϵ1/2 τ · n)∂Ω − (ϵ1/2 v, ∇ · τ) = (v, ϵ1/2 τ · n)Γ+ − (ϵ1/2 v, ∇ · τ)
+=
+(v, ϵ1/2 τ · n)Γ+ + (v, c v) − (v, ϵ1/2 ∇ · τ + β · ∇v + c v) + (v β, ∇v)
+and that
+(∇v, v β)
+=
+1
+2 ∥(β · n)1/2 v∥2
+0,Γ+ − 1
+2 (v, v ∇ · β) .
+Combining the above two equalities yields
+I =
+�
+v, ϵ1/2 ∇ · τ + β · ∇v + c v
+�
+− (v, ϵ1/2 τ · n)Γ+.
+(3.10)
+By the trace theorem and the Cauchy-Schwarz inequality, we have
+∥τ · n∥−1/2,Γ+
+≤
+C
+�
+∥τ∥ + ∥∇ · τ∥
+�
+≤
+C
+�
+G1/2
+1
+(τ, v; 0) + ∥ϵ1/2 ∇v∥ + ϵ−1/2 ∥β · ∇v∥ + ϵ−1/2 ∥c v∥
+�
+≤
+C ϵ−1/2�
+G1/2
+1
+(τ, v; 0) + ∥∇v∥ + ∥v∥
+�
+.
+(3.11)
+
+6
+Let αi = 1 for i = 2 or 1/2 for i = 3. Then it follows from (3.10), the Cauchy-Schwarz
+inequality, the definition of the dual norm, and (3.11) that for i = 2 and 3
+I
+≤
+∥v∥ ∥ϵ1/2 ∇ · τ + β · ∇v + c v∥ + ∥ϵ1/2−αi v∥1/2,Γ+ ∥ϵαi τ · n∥−1/2,Γ+
+(3.12)
+≤
+C
+�
+∥v∥ + ∥ϵαi τ · n∥−1/2,Γ+
+�
+G1/2
+i
+(τ, v; 0)
+≤
+C Gi(τ, v; 0) + C
+�
+∥ϵαi−1/2 ∇v∥ + ∥v∥
+�
+G1/2
+i
+(τ, v; 0),
+which, together with (3.9), implies
+∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ Ci Gi(τ, v; 0)
+(3.13)
+with C2 independent of ϵ and C3 proportional to ϵ−1/2. This completes the proof of (3.7)
+and, hence, (3.5) for i = 2 and 3.
+The validity of (3.5) for i = 1 may be established in a similar fashion by noticing that
+the boundary term of I in (3.8) vanishes due to the boundary conditions. This completes
+the proof of the theorem.
+4
+Mesh-dependent least-squares functionals
+For computational feasibility, in this section, we replace the 1
+2-norm in the least-squares
+functionals defined in (3.2) and (3.3) by mesh-dependent L2-norms. For the simplicity
+of presentation, assume that the domain Ω is a convex polygon in the two dimensional
+plane. (The extension to the higher dimension is straightforward.) Let Th = {K} be a
+triangulation of Ω with triangular elements K of diameter less than or equal to h. Assume
+that the triangulation Th is regular and quasi-uniform (see [18]).
+Denote by Eh the set of all edges of the triangulation Th. The least-squares functionals
+G2 and G3 defined in (3.2) and (3.3) are modified by the following computable least-squares
+functionals:
+Gh
+2(τ, v; f)
+=
+G1(τ, v; f) +
+�
+e∈Eh∩Γ+
+h−1
+e ∥ϵ−1/2 v∥2
+0,e
+(4.1)
+and
+Gh
+3(τ, v; f)
+=
+G1(τ, v; f) +
+�
+e∈Eh∩Γ+
+h−1
+e ∥v∥2
+0,e,
+(4.2)
+where he denotes the diameter of the edge e.
+For any triangle K ∈ Th, let Pk(K) be the space of polynomials of degree less than or
+equal to k on K and denote the local Raviart–Thomas space of index k on K by
+RTk(K) = Pk(K)2 +
+� x1
+x2
+�
+Pk(K).
+Then the standard H(div; Ω) conforming Raviart–Thomas space of index k [22] and the
+standard (conforming) continuous piecewise polynomials of degree k + 1 are defined, re-
+spectively, by
+Σk
+h = {τ ∈ H(div; Ω) : τ|K ∈ RTk(K), ∀ K ∈ Th},
+(4.3)
+V k+1
+h
+= {v ∈ H1(Ω) : v ∈ Pk+1(K), ∀ K ∈ Th}.
+(4.4)
+
+7
+These spaces have the following approximation properties: let k ≥ 0 be an integer, and
+let l ∈ (0, k + 1]:
+inf
+τ ∈ Σk
+h
+∥σ − τ∥H(div; Ω) ≤ C hl (∥σ∥l + ∥∇ · σ∥l)
+(4.5)
+for σ ∈ Hl(Ω)2 ∩ H(div; Ω) with ∇ · σ ∈ Hl(Ω) and
+inf
+v∈V k+1
+h
+∥u − v∥1 ≤ C hl ∥u∥l+1
+(4.6)
+for u ∈ Hl+1(Ω). In the subsequent sections, based on the smoothness of σ and u, we will
+choose k + 1 to be the smallest integer greater than or equal to l. Since the triangulation
+Th is regular, the following inverse inequalities hold for all K ∈ Th:
+∥τ∥1,K
+≤
+C h−1
+K ∥τ∥K,
+∀ τ ∈ RTk(K)
+(4.7)
+∥v∥1,K
+≤
+C h−1
+K ∥v∥K,
+∀ v ∈ Pk(K)
+(4.8)
+with positive constant C independent of hK.
+Denote by Uh
+i the finite dimensional subspaces of Ui:
+Uh
+i =
+�
+Σk
+h × V k+1
+h
+�
+∩ Ui.
+(4.9)
+For any (τ, v) ∈ Uh
+i , define the following norms:
+Mh
+2 (τ, v)
+=
+M1(τ, v) +
+�
+e∈Eh∩Γ+
+h−1
+e ∥ϵ−1/2 v∥2
+0,e
+and
+Mh
+3 (τ, v)
+=
+M1(τ, v) +
+�
+e∈Eh∩Γ+
+h−1
+e ∥v∥2
+0,e.
+Below we establish the discrete version of Theorem 3.1, i.e., the coercivity of the discrete
+functionals (4.1) and (4.2) with respect to the norms defined above. For the consistence
+of notation, we also let Gh
+1 = G1 and Mh
+1 = M1.
+Theorem 4.1. For all (τ, v) ∈ Uh
+i with i = 2 and 3, there exist positive constants Ci
+independent of ϵ such that
+Mh
+i (τ, v) ≤ Ci Gh
+i (τ, v; 0).
+(4.10)
+Proof. Similar to the argument in the proof of Theorem 3.1, in order to establish (4.10),
+it suffices to show that
+∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C Gh
+i (τ, v; 0)
+(4.11)
+for all (τ, v) ∈ Uh
+i . Moreover, we have
+∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C
+�
+Gh
+i (τ, v; 0) + I
+�
+(4.12)
+
+8
+with I defined in (3.8).
+For any e ∈ Eh ∩ Γ+, let e be an edge of element K ∈ Th. It follows from the trace
+theorem and the inverse inequality in (4.7) that
+he ∥τ · n∥2
+0,e ≤ C he ∥τ∥2
+0,e ≤ C he ∥τ∥0,K∥τ∥1,K ≤ C ∥τ∥2
+0,K,
+which, together with (3.6), implies
+�
+�
+�
+e∈Eh∩Γ+
+he ∥τ · n∥2
+0,e
+�
+�
+1/2
+≤ C ∥τ∥ ≤ C
+�
+G1/2
+1
+(τ, v; 0) + ∥ϵ1/2 ∇v∥
+�
+.
+(4.13)
+Let αi = 1 for i = 2 or 1/2 for i = 3. It follows from (3.10), the Cauchy-Schwarz
+inequality, and (4.13) that
+I
+=
+�
+v, ϵ1/2 ∇ · τ + β · ∇v + c v
+�
+− (v, ϵ1/2 τ · n)Γ+
+≤
+C
+�
+�∥v∥ + ϵαi
+�
+�
+e∈Eh∩Γ+
+he ∥τ · n∥2
+0,e
+�1/2
+�
+� Gh
+i (τ, v; 0)1/2
+≤
+C Gh
+i (τ, v; 0) + C
+�
+∥v∥ + ∥ϵ1/2∇v∥
+�
+Gh
+i (τ, v; 0)1/2
+which, together with (4.12), implies the validity of (4.11) and, hence, (4.10). This com-
+pletes the proof of the theorem.
+Remark 4.2. Note that the coercivity constant C3 in the discrete version is no longer
+depending on ϵ, that is better than the continuous version (see Theorem 3.1).
+5
+Finite element approximations
+The least-squares problems are to find (σ, u) ∈ Ui (i = 1, 2, 3) such that
+Gh
+i (σ, u; f) =
+min
+(τ , v)∈ Ui
+Gh
+i (τ, v; f).
+(5.1)
+The corresponding variational problems are to find (σ, u) ∈ Ui such that
+ai(σ, u; τ, v) = Fi(τ, v),
+∀ (τ, v) ∈ Ui,
+(5.2)
+where the bilinear forms ai(· ; ·) are symmetric and given by
+a1(σ, u; τ, v)
+=
+(σ + ϵ1/2 ∇u, τ + ϵ1/2 ∇v)
++(ϵ1/2 ∇ · σ + β · ∇u + c u, ϵ1/2 ∇ · τ + β · ∇v + c v),
+a2(σ, u; τ, v)
+=
+a1(σ, u; τ, v) +
+�
+e ∈Eh∩Γ+
+h−1
+e
+ϵ−1 (u, v)0,e,
+a3(σ, u; τ, v)
+=
+a1(σ, u; τ, v) +
+�
+e ∈Eh∩Γ+
+h−1
+e
+(u, v)0,e,
+
+9
+and the linear forms Fi(·) are given by
+Fi(τ, v) = (f, ϵ1/2 ∇ · τ + β · ∇v + c v)
+for i = 1, 2, 3.
+The least-squares finite element approximations to the variational problems in (5.2)
+are to find (σi
+h, ui
+h) ∈ Uh
+i such that
+ai(σi
+h, ui
+h; τ, v) = Fi(τ, v),
+∀ (τ, v) ∈ Uh
+i ,
+(5.3)
+for i = 1, 2, 3. Taking the difference between (5.2) and (5.3) implies the following orthog-
+onality:
+ai(σ − σi
+h, u − ui
+h; τ, v) = 0,
+∀ (τ, v) ∈ Uh
+i .
+(5.4)
+In the rest of this section, we consider a stronger norm which incorporates the norm
+of the streamline derivative:
+|||(τ, v)|||2
+i = Mh
+i (τ, v) +
+�
+K∈Th
+δK ∥β · ∇v∥2
+K,
+where δK is a positive constant to be determined. In the following lemma, we show that
+Gh
+i (σ, u; 0) are also elliptic with respect to these norms if the δK is appropriately chosen.
+Lemma 5.1. For all K ∈ Th, assume that 0 < δK ≤ min{h2
+K/ϵ, C}, then there exist
+positive constants Ci independent of ϵ such that
+|||(τ, v)|||2
+i ≤ Ci Gh
+i (τ, v; 0),
+∀ (τ, v) ∈ Uh
+i ,
+i = 1, 2, 3.
+Proof. By Theorems 3.1 and 4.1, to prove the validity of the lemma, it suffices to show
+that
+�
+K∈Th
+δK ∥β · ∇v∥2
+K ≤ Ci Gh
+i (τ, v; 0).
+(5.5)
+To this end, note the facts that
+δK ≤ C
+and
+δK ϵ
+h2
+K
+≤ min
+�
+1, C ϵ
+h2
+K
+�
+≤ C.
+Now it follows from the Cauchy-Schwarz inequality and the inverse inequality in (4.7) that
+�
+K∈Th
+δK ∥β · ∇v∥2
+K
+≤
+C
+�
+K∈Th
+δK
+�
+Gh
+1,K (τ, v; 0) + ∥ϵ1/2 ∇ · τ∥2
+K + ∥c v∥2
+K
+�
+≤
+C
+�
+K∈Th
+�
+Gh
+1,K (τ, v; 0) + δK ϵ
+h2
+K
+∥τ∥2
+K + ∥v∥2
+K
+�
+≤
+C
+�
+Gh
+1 (τ, v; 0) + ∥τ∥2 + ∥v∥2�
+≤ C Gh
+i (τ, v; 0),
+which establishes (5.5) and hence completes the proof of the lemma.
+
+10
+To choose δK properly, first define the local mesh P´eclet number by
+PeK = ∥β∥0,∞,K hK
+2 ϵ
+,
+then partition the triangulation Th into two subsets:
+T c
+h = {K ∈ Th : PeK > 1}
+and
+T d
+h = {K ∈ Th : PeK ≤ 1}.
+(5.6)
+The elements in T c
+h are referred to the convection-dominated elements, while the elements
+in T d
+h the diffusion-dominated elements. Now, the δK is chosen to be
+δK =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2 hK
+∥β∥0,∞,K
+,
+if K ∈ T c
+h ,
+h2
+K
+ϵ ,
+if K ∈ T d
+h .
+(5.7)
+Remark 5.2. The δK defined in (5.7) satisfies the assumption in Lemma 5.1, i.e.,
+δK ≤ min{h2
+K/ϵ, C}.
+(5.8)
+Proof. Since ∥β∥0,∞,K is large comparing to hK, we have
+2 hK
+∥β∥0,∞,K
+≤ C.
+(5.9)
+For any K ∈ T c
+h , the fact that PeK > 1 implies
+2 hK
+∥β∥0,∞,K
+< h2
+K
+ϵ ,
+which, together with (5.9), yields (5.8). For any K ∈ T d
+h , (5.8) is again a consequence of
+the definition of δK in (5.7), the fact that PeK ≤ 1, and (5.9).
+Denote by T ∂
+h the set of elements that intersect the outflow boundary nontrivially, i.e.,
+T ∂
+h = {K ∈ Th : meas( ¯K ∩ Γ+) > 0}.
+In this paper, we assume that
+T ∂
+h ⊂ T d
+h .
+(5.10)
+For any K ∈ T d
+h , the fact that PeK ≤ 1 implies
+hK <
+2 ϵ
+∥β∥0,∞,K
+.
+Hence, assumption (5.10) means that the mesh size in the boundary layer region is com-
+parable to the perturbation parameter ϵ.
+
+11
+Theorem 5.3. Let (σ, u) be the solution of (5.2). Assume that (σ, u) ∈ Hl(Ω)2×Hl+1(Ω)
+and that ∇ · σ ∈ Hl(Ω). Let (σi
+h, ui
+h), i = 1, 2, 3, be the solution of (5.3) with k = l.
+Under the assumption in (5.10), we have the following a priori error estimation:
+Ci
+������(σ − σi
+h, u − ui
+h)
+������2
+i
+≤
+�
+K∈T c
+h
+h2l−1
+K
+�
+ϵ ∥∇ · σ∥2
+l,K + hK ∥σ∥2
+l,K + ∥u∥2
+l+1,K
+�
++
+�
+K∈T d
+h
+h2l−1
+K
+� ϵ2
+hK
+∥∇ · σ∥2
+l,K + hK ∥σ∥2
+l,K + ϵ
+hK
+∥u∥2
+l+1,K
+�
+,
+(5.11)
+where constants Ci > 0 are independent of ϵ.
+Proof. We provide proof of (5.11) only for i = 2 and 3 since (5.11) may be obtained in a
+similar fashion.
+To this end, let σI and uI be the interpolants of σ and u, respectively, such that the
+approximation properties in (4.5) and (4.6) hold and that
+(∇ · (σ − σI), v) = 0,
+∀ v ∈ Dh
+k,
+(5.12)
+where Dh
+k = {v ∈ L2(Ω) : v|K ∈ Pk(K) ∀ K ∈ Th} is the space of discontinuous piecewise
+polynomials of degree less than or equal to k ≥ 0. Let
+EI = σ − σI,
+Ei
+h = σI − σi
+h,
+eI = u − uI,
+and
+ei
+h = uI − ui
+h.
+Since Ei = σ − σi
+h = EI + Ei
+h and ei = u − ui
+h = eI + ei
+h, the triangle inequality gives
+������(Ei, ei)
+������
+i ≤ |||(EI, eI)|||i +
+������(Ei
+h, ei
+h)
+������
+i.
+(5.13)
+Let αi = −1 or 0 for i = 2, 3. By approximation property (4.6) and assumption (5.10),
+we have
+�
+e ∈Eh∩Γ+
+h−1
+e
+ϵαi ∥eI∥2
+0,e ≤ C
+�
+K∈T ∂
+h
+h2l
+K ϵαi ∥u∥2
+l+1,K ≤ C
+�
+K∈T ∂
+h
+h2l+αi
+K
+∥u∥2
+l+1,K.
+Now, it follows from (4.5), (4.6), the trace theorem, and the fact δK ≤ C that
+|||(EI, eI)|||2
+i
+≤
+C
+�
+�∥EI∥2 + ∥eI∥2 + ∥ϵ1/2 ∇eI∥2 +
+�
+e∈Γ+
+h−1
+e
+ϵαi ∥eI∥2
+e +
+�
+K∈Th
+∥β · ∇eI∥2
+K
+�
+�
+≤
+C
+�
+� �
+K∈Th
+h2l
+K ∥σ∥2
+l,K +
+�
+K∈Th
+h2l
+K ∥u∥2
+l+1,K +
+�
+K∈T ∂
+h
+h2l+αi
+K
+∥u∥2
+l+1,K
+�
+� .
+(5.14)
+
+12
+To bound the second term of the right-hand side in (5.13), by Lemma 5.1 and orthog-
+onality (5.4), we have
+Ci
+������(Ei
+h, ei
+h)
+������2
+i ≤ ai(Ei
+h, ei
+h; Ei
+h, ei
+h) = ai(Ei
+h, ei
+h; −EI, −eI) ≡ Ii
+1 + Ii
+2 + Ii
+3 + Ii
+4, (5.15)
+where
+Ii
+1
+=
+(c ei
+h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI) + (Ei
+h + ϵ1/2 ∇ei
+h, −EI − ϵ1/2 ∇eI),
+Ii
+2
+=
+(ϵ1/2 ∇ · Ei
+h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI),
+Ii
+3
+=
+(β · ∇ei
+h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI),
+and
+Ii
+4
+=
+�
+e ∈Eh∩Γ+
+h−1
+e
+ϵαi (ei
+h, −eI)0,e.
+It follows from the triangle and Cauchy-Schwarz inequalities, (4.5), and (4.6) that
+Ii
+1
+≤ C ∥ei
+h∥
+�
+∥ϵ1/2∇ · EI∥ + ∥∇eI∥ + ∥eI∥
+�
++ C
+�
+∥Ei
+h∥ + ∥ϵ1/2∇ei
+h∥
+� �
+∥EI∥ + ∥ϵ1/2∇eI)∥
+�
+≤C
+�
+∥ei
+h∥ + ∥Ei
+h∥ + ∥ϵ1/2∇ei
+h∥
+�
+�
+� �
+K∈Th
+h2l
+K
+�
+ϵ∥∇ · σ∥2
+l,K + ∥σ∥2
+l,K + ∥u∥2
+l+1,K
+�
+�
+�
+1/2
+. (5.16)
+By (5.12), the Cauchy-Schwarz and triangle inequalities, and the inverse inequality in
+(4.7), we have
+Ii
+2 = −(ϵ1/2 ∇ · Ei
+h, β · ∇eI + c eI),
+≤ C
+�
+K∈Th
+ϵ1/2
+hK
+∥Ei
+h∥K
+�
+∥∇eI∥K + ∥eI∥K
+�
+≤ C ∥Ei
+h∥
+�
+� �
+K∈Th
+ϵ h2l−2
+K
+∥u∥2
+l+1,K
+�
+�
+1/2
+. (5.17)
+By the Cauchy-Schwarz and the triangle inequalities, I3 is bounded by
+Ii
+3 ≤ C
+�
+K∈Th
+∥β · ∇ei
+h∥K
+�
+ϵ1/2 ∥∇ · EI∥K + ∥∇eI∥K + ∥eI∥K
+�
+≤
+C
+�
+K∈Th
+∥β · ∇ei
+h∥K
+�
+ϵ1/2 hl
+K ∥∇ · σ∥l,K + hl
+K ∥u∥l+1,K
+�
+≤ C
+�
+� �
+K∈Th
+δK∥β · ∇ei
+h∥2
+K
+�
+�
+1/2�
+� �
+K∈Th
+δ−1
+K
+�
+ϵ h2l
+K ∥∇ · σ∥2
+l,K + h2l
+K ∥u∥2
+l+1,K
+�
+�
+�
+1/2
+.(5.18)
+
+13
+For Ii
+4, it follows from the Cauchy-Schwarz inequality and the trace theorem that
+Ii
+4
+≤
+C
+�
+�
+�
+e ∈Eh∩Γ+
+h−1
+e
+ϵαi ∥ei
+h∥2
+0,e
+�
+�
+1/2 �
+�
+�
+e ∈Eh∩Γ+
+h−1
+e
+ϵαi ∥eI∥2
+0,e
+�
+�
+1/2
+≤
+C
+�
+�
+�
+e ∈Eh∩Γ+
+h−1
+e
+ϵαi ∥ei
+h∥2
+0,e
+�
+�
+1/2 �
+� �
+K∈T ∂
+h
+h2l+αi
+K
+∥u∥2
+l+1,K
+�
+�
+1/2
+.
+(5.19)
+Combining (5.15), (5.16), (5.17), (5.18), (5.19), and (5.8), we have
+Ci
+������(Ei
+h, ei
+h)
+������2
+i
+≤
+�
+K∈Th
+h2l
+K∥σ∥2
+l,K +
+�
+K∈Th
+�
+1 + δ−1
+K
+�
+ϵ h2l
+K ∥∇ · σ∥2
+l,K +
+�
+K∈T ∂
+h
+h2l+αi
+K
+∥u∥2
+l+1,K
++
+�
+K∈Th
+�
+1 + ϵ h−2
+K + δ−1
+K
+�
+h2l
+K∥u∥2
+l+1,K
+≤
+�
+K∈Th
+�ϵ h2l
+K
+δK
+∥∇ · σ∥2
+l,K + h2l
+K ∥σ∥2
+l,K + h2l
+K
+δK
+∥u∥2
+l+1,K
+�
++
+�
+K∈T ∂
+h
+h2l+αi
+K
+∥u∥2
+l+1,K,
+which, together with the definition of δK in (5.7), implies
+Ci
+������(Ei
+h, ei
+h)
+������2
+i
+≤
+�
+K∈T c
+h
+h2l−1
+K
+�
+ϵ ∥∇ · σ∥2
+l,K + hK ∥σ∥2
+l,K + ∥u∥2
+l+1,K
+�
++
+�
+K∈T d
+h
+h2l−1
+K
+� ϵ2
+hK
+∥∇ · σ∥2
+l,K + hK ∥σ∥2
+l,K + ϵ
+hK
+∥u∥2
+l+1,K
+�
+.
+Now, (5.11) is a consequence of (5.13) and (5.14). This completes the proof of the theorem.
+Note that the a priori error estimate in Theorem 5.3 is not optimal. This is because
+the coercivity of the homogeneous least-squares functionals in Lemma 5.1 are established
+in a norm that is weaker than the norm used for the continuity of the functionals. To
+restore the full order of convergence, one may use piecewise polynomials of degree l + 1 to
+approximate u.
+Theorem 5.4. Let (σi
+h, ui
+h), i = 1, 2, 3, be the solution of (5.3) with Uh
+i = (Σl
+h×V l+1
+h
+)∩ Ui.
+
+14
+Under the assumption of Theorem 5.3, we have the following a priori error estimation:
+Ci
+������(σ − σi
+h, u − ui
+h)
+������2
+i
+≤
+�
+K∈T c
+h
+h2l
+K
+�
+∥∇ · σ∥2
+l,K + ∥σ∥2
+l,K + hK ∥u∥2
+l+2,K
+�
++
+�
+K∈T d
+h
+h2l
+K
+� ϵ2
+h2
+K
+∥∇ · σ∥2
+l,K + ∥σ∥2
+l,K + ϵ ∥u∥2
+l+2,K
+�
+,
+(5.20)
+where constants Ci > 0 are independent of ϵ.
+Proof. The a priori error estimate in (5.20) may be obtained in a similar fashion by noting
+that
+∥u − uI∥1 ≤ C hl+1∥u∥l+2.
+6
+Adaptive algorithm
+Asymptotic analysis (see, e.g., [20]) shows that the solution of a convection-dominated
+diffusion-reaction problem consists of two parts: the solution of the reduced equation
+(ϵ = 0) and the correction, i.e., the boundary or interior layers.
+The boundary and
+interior layers are narrow regions where derivatives of the solution change dramatically.
+For example, for the following problem [20]:
+�
+�
+�
+�
+�
+−ϵ ∆u + ∂u
+∂y = f
+in Ω = (0, 1)2,
+u = 0
+on ∂Ω,
+the exponential layer is of width O(ϵ) at y = 1, and the width of the parabolic boundary
+layers is O(ϵ1/2) at both x = 0 and x = 1.
+Therefore, two sets of largely different
+scales exist simultaneously in the convection-dominated diffusion problem, and hence it is
+difficult computationally.
+On the one hand, one can apply the small scale over the entire domain, i.e., to use
+uniform fine meshes. With such a fine mesh, the standard Galerkin finite element method
+can also produce a good approximation. However, it is computationally inefficient due to
+the small region of the boundary and/or interior layers. On the other hand, one can use
+the large scale over the entire domain. If the outflow boundary conditions are imposed
+strongly, the numerical solution (away from the boundary layers) will be polluted. In
+contrast, if the outflow boundary conditions are imposed weakly, the boundary layers can
+not be resolved (see, e.g., numerical results in [4, 17]).
+Neither of the above two approaches leads to a satisfactory numerical scheme. The fail-
+ure is due to the fact that these approaches ignore this intrinsic property of the convection-
+dominated diffusion problem. In contrast, the Shishkin mesh is aware of and respect it.
+
+15
+Basically, the Shishkin mesh is a piecewise uniform mesh. In the diffusion-dominated re-
+gion where the layers stand, it is a fine mesh suitable to the layer and in the convective
+region, it turns to be a coarse mesh. The disadvantage of the Shishkin mesh is that it
+needs the a priori information of the solution, such as the location and the width of the
+layer, in order to construct a mesh of high quality. However, this information is not always
+available in advance, especially, for a complex problem.
+Based on the above considerations, we employ adaptive least-squares finite element
+methods. The least-squares estimators are simply defined as the value of the least-squares
+functionals at the current approximation. To this end, for each element K ∈ Th, denote
+the local least-squares functionals by
+Gh
+1,K(τ, v; f)
+=
+∥τ + ϵ1/2 ∇v∥2
+K + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2
+K,
+Gh
+2,K(τ, v; f)
+=
+�
+�
+�
+�
+�
+Gh
+1,K(τ, v; f),
+if K ∩ Γ+ = ∅,
+Gh
+1,K(τ, v; f) +
+�
+e∈K∩Γ+
+h−1
+e ∥ϵ−1/2v∥2
+0, e,
+otherwise,
+and Gh
+3,K(τ, v; f)
+=
+�
+�
+�
+�
+�
+Gh
+1,K(τ, v; f),
+if K ∩ Γ+ = ∅,
+Gh
+1,K(τ, v; f) +
+�
+e∈K∩Γ+
+h−1
+e ∥v∥2
+0, e,
+otherwise.
+Let (ˆσh
+i , ˆuh
+i ) be the current approximations to the solutions of (5.3) for i = 1, 2, 3. Then
+the least-squares indicators are simply the square root of the value of the local least-squares
+functionals at the current approximation:
+ηi
+K = Gh
+i,K (ˆσi
+h, ˆui
+h; f)1/2
+(6.1)
+for all K ∈ Th and for i = 1, 2, 3. The least-squares estimators are
+ηi =
+�
+� �
+K∈Th
+�
+ηi
+K
+�2
+�
+�
+1/2
+= Gh
+i (ˆσi
+h, ˆui
+h; f)1/2
+(6.2)
+for i = 1, 2, 3.
+Let (σ, u) be the solution of (5.2) and denote the true errors by
+ˆEi = σ − ˆσi
+h
+and
+ˆei = u − ˆu1
+h
+for
+i = 1, 2, 3.
+Theorem 6.1. There exist positive constants Ce,1 and Cr,1 independent of ϵ such that
+η1
+K ≤ Ce,1
+�
+M1,K(ˆE1, ˆe1) + ∥β · ∇ ˆe1∥2
+K + ϵ ∥∇ · ˆE1∥2
+K
+�1/2
+(6.3)
+for all K ∈ T and that
+M1(ˆE1, ˆe1)1/2 ≤ Cr,1 η1.
+(6.4)
+
+16
+Proof. Since the exact solution (σ, u) satisfies (2.4), we have
+�
+η1
+K
+�2 = Gh
+1,K(ˆE1, ˆe1; 0)
+and
+�
+η1�2 = Gh
+1(ˆE1, ˆe1; 0).
+which, together with the triangle inequality and Theorem 3.1, imply the efficiency and the
+reliability bounds, respectively.
+Theorem 6.2. There exist positive constants Ce,i independent of ϵ such that
+Ce, i
+�
+ηi
+K
+�2 ≤ Mh
+i,K(ˆEi, ˆei) + ∥β · ∇ˆei∥2
+K + ϵ ∥∇ · ˆEi∥2
+(6.5)
+for all K ∈ T and i = 2, 3.
+Proof. Let αi = −1 for i = 2 or 0 for i = 3. With the fact that (σ, u) is the exact solution
+satisfying (2.4), we have
+ηi(ˆσh
+i , ˆuh
+i )2 = Gh
+i (ˆσh
+i , ˆuh
+i ; f)
+=
+∥ˆσh
+i + ϵ1/2 ∇ˆuh
+i ∥2 + ∥ϵ1/2 ∇ · ˆσh
+i + β · ∇ˆuh
+i + c ˆuh
+i − f∥2 +
+�
+e∈Eh∩Γ+
+ϵαi h−1
+e ∥ˆuh
+i ∥2
+0,e
+=
+∥ˆEi + ϵ1/2 ∇ˆei∥2 + ∥ϵ1/2 ∇ · ˆEi + β · ∇ˆei + c ˆei∥2 +
+�
+e∈Eh∩Γ+
+ϵαi h−1
+e ∥ˆei∥
+=
+Gh
+i (ˆEi, ˆei; 0),
+(6.6)
+with which, the efficiency bound simply follows from (6.6) and the Cauchy-Schwarz in-
+equality.
+In the remainder of this section, we describe the standard adaptive mesh refinement
+algorithm. Starting with an initial triangulation T0, a sequence of nested triangulations
+{Tl} is generated through the well known AFEM-Loop:
+SOLVE −→ ESTIMATE −→ MARK −→ REFINE.
+The SOLVE step solves (5.3) in the finite element space corresponding to the mesh
+Tl for a numerical approximation (σi
+h(l), ui
+h(l)) ∈ Uh
+i (l), where Uh
+i (l) is the finite element
+space defined on Tl. Hereafter, we shall explicitly express the dependence of a quantity
+on the level l by either the subscript like Tl or the variable like Uh
+i (l).
+The ESTIMATE step computes the indicators {ηi
+K(l)} and the estimator ηi(l) defined
+in (6.1) and (6.2), respectively.
+The way to choose elements for refinement influences the efficiency of the adaptive
+algorithm. If most of elements are marked for refinement, then it is comparable to uniform
+refinement, which does not take full advantage of the adaptive algorithm and results in
+redundant degrees of freedom. On the other hand, if few elements are refined, then it
+requires many iterations, which undermines the efficiency of the adaptive algorithm, since
+each iteration is costly. For the singularly perturbed problems, it is well known that the
+indicators associated with the elements in the layer region are much larger than others.
+
+17
+Therefore, we MARK by the maximum algorithm, which defines the set ˆTl of marked
+elements such that for all K ∈ ˆTl
+ηi
+K(l) ≥ θ max
+K∈Tl ηi
+K(l).
+The REFINE step is to bisect all the triangles in ˆTl into two sub-triangles to generate
+a new triangulation Tl+1. Note that some triangles in Tl \ ˆTl adjacent to triangles in ˆTl are
+also refined in order to avoid hanging nodes.
+In summary, the adaptive least-squares finite element algorithm can be cast as follows:
+with the initial mesh T0, marking parameter θ ∈ (0, 1), and the maximal number of
+iteration maxIt, for l = 0, 1, · · · , maxIt, do
+(1) (σi
+h(l), ui
+h(l)) = SOLVE(Tl);
+(2) {ηi
+K(l)} = ESTIMATE(Tl, σi
+h(l), ui
+h(l));
+(3) ˆTl = MARK(Tl, {ηi
+K(l)});
+(4) Tl+1 = REFINE(Tl, ˆTl).
+7
+Numerical experiments
+In this section, we conduct several numerical experiments on two model problems used by
+many authors (see, e.g., [4, 17]). Both the model problems are defined in the unit square
+and all numerical experiments are started with the same initial mesh, which consists of
+sixteen isosceles right triangles. The marking parameter θ is chosen to be 0.6.
+7.1
+Boundary layer
+In this example, β = [1, 1]T , and c = 0, and the external force f is chosen such that the
+exact solution is
+u(x, y) = sin πx
+2 + sin πy
+2
+�
+1 − sin πx
+2
+�
++ e−1/ϵ − e−(1−x)(1−y)/ϵ
+1 − e−1/ϵ
+.
+This solution is smooth, but develops boundary layers at x = 1 and y = 1 with width
+O(ϵ). This example is suitable for testing capability of the numerical approximations on
+resolving the boundary layers.
+In this numerical experiment, ϵ = 10−3. Given the tolerance tol = 0.5, computation is
+terminated if
+ηi(l) ≤ tol.
+(7.1)
+Since the exact solution is available, the true error is computed and the effectivity index
+is defined as follows:
+eff-index :=
+ηi(σi
+h, ui
+h)
+������(σ − σi
+h, u − ui
+h)
+������
+i
+.
+(7.2)
+
+18
+Figure 1: The final meshes and the numerical solutions are, respectively, displayed in the
+first and the second columns and the rows are corresponding to i = 1, 2, 3.
+The final meshes are displayed in the first column of Figure 1 when the stopping criterion
+(7.1) is satisfied. They clearly show that the refinements cluster around the boundary
+layer area. The numerical solutions on the final meshes are depicted in the second column
+of Figure 1. All the three methods successfully capture the sharp boundary layers, and
+no visible oscillation appears in the numerical solutions.
+Reported in Figure 2 is the
+
+0.9 .
+0.5
+0.1 .
+0.2 ..
+A
+:
+0.6
+: -?
+0.8
+0.4
+0.b
+0.2
+...
+0.4
+c.2
+00.5 .
+0.5 .
+0.4 .
+0.2
+0.2
+0.6
+0.8
+0.4
+0.6
+0.2
+0.4
+c.2
+00.5 .
+0.5 .
+0.4 .
+0.2
+5
+0.2
+0.6
+0.8
+0.4
+0.6
+0.2
+0.4
+c.2
+019
+convergence rates of the numerical solutions. The errors with the norm |||·|||i that are
+used in the a priori error estimate are tracked, which converge in the order of (DoF)−1.
+Moreover, the convergence rate is independent of the value of ϵ. This is also verified by
+the test problem with ϵ = 10−4, where the convergence rate does not deteriorate (see the
+second column of Figure 2).
+Figure 2: The convergence rates corresponding to ϵ = 10−3 and 10−4 are displayed in the
+first and the second columns, respectively, and the rows are corresponding to i = 1, 2, 3.
+
+10°
+10°
+10
+10°
+10
+10*
+10°
+errEne3
+estimator
+DoF-1
+effindex
+10
+102
+103
+104
+105
+10°
+Degree of Freedom10°
+10
+00.000
+10°
+10~
+10°
+10
+10
+errEne
+estimator
+10
+DoF-1
+effindex
+10°
+102
+103
+104
+105
+10°
+Degree of Freedom10°
+102
+10°
+DODO
+00:
+080:00
+10°
+10
+10*
+10°
+10
+errEne
+estimator
+10
+DoF-1
+effindex
+10
+102
+103
+104
+105
+10°
+Degree of Freedom10
+10
+0000::
+80:8
+10
+10~
+10*
+10
+10
+erEne
+estimator
+10
+DoF-1
+effindex
+10
+102
+103
+104
+105
+10°
+Degree of Freedom10
+10
+10
+10°
+10
+10°
+10
+errEne
+estimator
+10
+DoF-1
+effindex
+0
+10
+102
+103
+10*
+105
+10°
+Degree of Freedom10°
+10
+：*黑
+%08
+00:000
+10°
+10~
+10*
+10
+10
+errEne
+estimator
+10
+DoF-1
+effindex
+10°
+102
+103
+104
+105
+10°
+Degree of Freedom20
+7.2
+Interior layer
+In the second example, β = [1/2,
+√
+3/2]T , c = 0, f = 0, and the boundary condition is
+u =
+�
+�
+�
+�
+�
+�
+�
+1,
+on {(x, y) : y = 0, 0 ≤ x ≤ 1},
+1,
+on {(x, y) : x = 0, 0 ≤ y ≤ 1/5},
+0,
+otherwise.
+The exact solution of the problem remains unknown. However, it is known that, additional
+to the boundary layers, the solution develops an interior layer along the line y =
+√
+3 x+0.2
+due to the discontinuity at (0, 0.2) of the boundary condition. The problem is chosen to
+test whether the formulations can capture the interior layers.
+Figure 3 shows that all the three methods capture both the boundary and the interior
+layers. Moreover, the numerical solutions do not exhibit any visible oscillation, which is
+much better than the results reported in [4].
+Figure 3: Numerical solutions corresponding to i = 1, 2, 3 from left to right.
+Acknowledgements
+We thank Dr. Shuhao Cao for the discussion and helpful suggestions on the computation
+of the test problems.
+References
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+1
+0.6
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+0.b
+0.2
+0.4
+c.21.2
+0.9 .
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+0.6
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+0.4
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+

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+arXiv:2301.03762v1  [math.AG]  10 Jan 2023
+REGULAR SEMISIMPLE HESSENBERG VARIETIES WITH COHOMOLOGY RINGS
+GENERATED IN DEGREE TWO
+MIKIYA MASUDA AND TAKASHI SATO
+Abstract. A regular semisimple Hessenberg variety Hess(S, h) is a smooth subvariety of the flag variety
+determined by a square matrix S with distinct eigenvalues and a Hessenberg function h. The cohomology
+ring H∗(Hess(S, h)) is independent of the choice of S and is not explicitly described except for a few cases.
+In this paper, we characterize the Hessenberg function h such that H∗(Hess(S, h)) is generated in degree two
+as a ring. It turns out that such h is what is called a (double) lollipop.
+1. Introduction
+The flag variety Fl(n) consists of nested sequences of linear subspaces in the complex vector space Cn:
+Fl(n) = {V• = (V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn) | dimC Vi = i
+(∀i ∈ [n] = {1, 2, . . ., n})}.
+A Hessenberg function h: [n] → [n] is a monotonically non-decreasing function satisfying h(j) ≥ j for any
+j ∈ [n]. We often express a Hessenberg function h as a vector (h(1), . . . , h(n)) by listing the values of h.
+Given an n × n matrix A and a Hessenberg function h, a Hessenberg variety Hess(A, h) is defined as
+Hess(A, h) = {V• ∈ Fl(n) | AVi ⊂ Vh(i)
+(∀i ∈ [n])}
+where the matrix A is regarded as a linear operator on Cn. Note that Hess(A, h) = Fl(n) if h = (n, . . . , n).
+The family of Hessenberg varieties Hess(A, h) contains important varieties such as Springer fibers (A is
+nilpotent and h = (1, 2, . . ., n)), Peterson varieties (A is regular nilpotent and h = (2, 3, . . . , n, n)), and
+permutohedral varieties (A is regular semisimple and h = (2, 3, . . . , n, n)), which are toric varieties with
+permutohedra as moment polytopes.
+Among n × n matrices, regular semisimple ones S (i.e. matrices S having distinct eigenvalues) are generic
+and Hess(S, h) is called a regular semisimple Hessenberg variety. The regular semisimple Hessenberg variety
+Hess(S, h) has nice properties. For instance, it is smooth and its cohomology H∗(Hess(S, h)) becomes a
+module over the symmetric group Sn on [n] by Tymoczko’s dot action [20]. Remarkably, the solution of
+Shareshian–Wachs conjecture [18] by Brosnan and Chow [5] (and Guay-Paquet [10]) connected H∗(Hess(S, h))
+as an Sn-module and chromatic symmetric functions on certain graphs. This opened a way to prove the
+famous Stanley–Stembridge conjecture in graph theory through the geometry or topology of Hessenberg
+varieties and motivated us to study H∗(Hess(S, h)). Note that H∗(Hess(S, h)) (indeed the diffeomorphism
+type of Hess(S, h)) is independent of the choice of S. We write the regular semisimple Hessenberg variety
+Hess(S, h) as X(h) for brevity since our concern in this paper is its cohomology ring.
+The Sn-module structure on H∗(X(h)) is determined in some cases (e.g. [12]). In particular, that on
+H2(X(h)) was explicitly described by Chow [7] combinatorially (through the theorem by Brosnan-Chow
+mentioned above) and by Cho-Hong-Lee [6] geometrically. Motivated by their works, Ayzenberg and the
+authors [4] reproved their results by giving explicit additive generators of H2(X(h)) in terms of GKM theory.
+The ring structure on H∗(X(h)) is not explicitly described except for a few cases. Remember that X(h)
+for h = (n, . . . , n) is the flag variety Fl(n) and H∗(Fl(n)) is generated in degree 2 as a ring. Moreover, X(h)
+for h = (2, 3, . . . , n, n) is the permutohedral variety and H∗(X(h)) is also generated in degree 2 as a ring. On
+the other hand, for h = (h(1), n, . . . , n) with h(1) arbitrary, a result of [2] shows that H∗(X(h)) is generated
+Date: January 11, 2023.
+2020 Mathematics Subject Classification. Primary: 57S12, Secondary: 14M15.
+Key words and phrases. Hessenberg variety, torus action, GKM theory, equivariant cohomology, lollipop.
+1
+
+2
+M. MASUDA AND T. SATO
+in degree 2 as a ring if and only if h(1) = 2 or n, where X(h) = Fl(n) for the latter case. Therefore, it is
+natural to ask when H∗(X(h)) is generated in degree 2 as a ring. The answer is the following, which is our
+main result in this paper.
+Theorem 1.1. Assume that h(j) ≥ j + 1 for j ∈ [n − 1]. Then H∗(X(h)) is generated in degree 2 as a ring
+if and only if h is of the following form (1.1) for some 1 ≤ a < b ≤ n,
+(1.1)
+h(j) =
+
+
+
+
+
+a + 1
+(1 ≤ j ≤ a)
+j + 1
+(a < j < b)
+n
+(b ≤ j ≤ n).
+Remark 1.1.
+(1) X(h) is connected if and only if h(j) ≥ j + 1 for any j ∈ [n − 1]. When X(h) is not
+connected, each connected component of X(h) is a product of smaller regular semisimple Hessenberg
+varieties.
+(2) X(h) is the flag variety Fl(n) when (a, b) = (n − 1, n) and is the permutohedral variety when (a, b) =
+(1, n).
+(3) We will give an explicit presentation of the ring structure on H∗(X(h)) for h of the form (1.1) in a
+forthcoming paper [17].
+We can visualize a Hessenberg function h by drawing a configuration of the shaded boxes on a square
+grid of size n × n, which consists of boxes in the i-th row and the j-th column satisfying i ≤ h(j). Since
+h(j) ≥ j for any j ∈ [n], the essential part is the shaded boxes below the diagonal. For example, Figure
+1 below is the configurations of two Hessenberg functions h of the form (1.1) with n = 11: one is h =
+(2, 3, 4, 5, 6, 7, 11, 11, 11, 11) where (a, b) = (1, 7) and the other is h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) where
+(a, b) = (3, 7). We often identify a Hessenberg function h with its configuration.
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+❅❅
+Figure 1. The configurations for h = (2, 3, 4, 5, 6, 7, 11, 11, 11, 11) and h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11)
+The chromatic symmetric functions and LLT polynomials associated with h of the form (1.1) are studied
+from the viewpoint of combinatorics in [8, 13], and when a = 1 or b = n, the corresponding Hessenberg
+functions
+h = (2, 3, . . . , b, n, . . ., n)
+or
+(a + 1, . . . , a + 1, a + 2, . . . , n − 1, n, n)
+are called lollipops in those papers, so the Hessenberg function of the form (1.1) may be called a double
+lollipop.
+The paper is organized as follows.
+In Section 2, we review GKM theory to compute the equivariant
+cohomology of X(h).
+We prove the “only if” part in Theorem 1.1 in Section 3.
+Indeed, we consider a
+Morse-Bott function fh on X(h), where the inverse image of the minimum or maximum value of fh is a
+regular semisimple Hessenberg variety X(h′) with h′ of size one less than that of h. Then a property of the
+
+REGULAR SEMISIMPLE HESSENBERG VARIETIES
+3
+Morse-Bott function fh shows the surjectivity of the restriction map H∗(X(h); Q) → H∗(X(h′); Q), and this
+enables us to use an inductive argument to prove the “only if” part. In Section 4, we prove the “if” part in
+Theorem 1.1 by applying the method developed in [2, 9] together with the explicit generators of H2(X(h))
+obtained in our previous work [4].
+2. Regular semisimple Hessenberg varieties
+We first recall some properties of a regular semisimple Hessenberg variety X(h).
+Theorem 2.1 ([14]).
+(1) X(h) is smooth.
+(2) dimC X(h) = �n
+j=1(h(j) − j).
+(3) X(h) is connected if and only if h(j) ≥ j + 1 for ∀j ∈ [n − 1].
+(4) Hodd(X(h)) = 0 and the 2k-th Betti number of X(h) is given by
+#{w ∈ Sn | ℓh(w) = k}
+where
+(2.1)
+ℓh(w) = #{1 ≤ j < i ≤ n | w(j) > w(i), i ≤ h(j)}.
+For calculation of the cohomology ring of X(h), we use equivariant cohomology which we shall explain. We
+assume that the matrix S in X(h) = Hess(S, h) is a diagonal matrix. Let T be an algebraic torus consisting
+of diagonal matrices in the general linear group GLn(C). The linear action of T on Cn induces an action on
+the flag variety Fl(n) and preserves X(h) since S commutes with T . The fixed point sets of the T -actions on
+X(h) and Fl(n) consist of all permutation flags, that is,
+(2.2)
+X(h)T = Fl(n)T ∼= Sn.
+Since T can naturally be identified with (C∗)n, the classifying space BT of T is B(C∗)n = (CP ∞)n.
+Let pi : T → C∗ be the projection on the i-th diagonal component of T and ti = p∗
+i (t) ∈ H2(BT ) where
+p∗
+i : H∗(BC∗) → H∗(BT ) and t ∈ H2(BC∗) is the first Chern class of the tautological line bundle over
+BC∗ = CP ∞. Then
+(2.3)
+H∗(BT ) = Z[t1, . . . , tn].
+The equivariant cohomology of the T -variety X(h) is defined as
+H∗
+T (X(h)) := H∗(ET ×T X(h))
+where ET is the total space of the universal principal T -bundle ET → BT and ET ×T X(h) is the orbit
+space of the product ET × X(h) by the diagonal T -action. The projection ET × X(h) → ET on the first
+factor induces a fibration
+X(h)
+ρ−→ ET ×T X(h)
+π−→ BT.
+Since Hodd(X(h)) = 0 as in Theorem 2.1 and Hodd(BT ) = 0, the Serre spectral sequence of the fibration above
+collapses. It implies that ρ∗ : H∗
+T (X(h)) → H∗(X(h)) is surjective and induces a graded ring isomorphism
+(2.4)
+H∗(X(h)) ∼= H∗
+T (X(h))/(π∗(t1), . . . , π∗(tn))
+by (2.3). Therefore, one can find the ring structure on H∗(X(h)) through H∗
+T (X(h)).
+Since Hodd(X(h)) = 0, it follows from the localization theorem that the homomorphism
+(2.5)
+H∗
+T (X(h)) → H∗
+T (X(h)T ) =
+�
+w∈Sn
+H∗
+T (w) =
+�
+w∈Sn
+Z[t1, . . . , tn] = Map(Sn, Z[t1, . . . , tn])
+induced from the inclusion map X(h)T → X(h) is injective, where X(h)T is identified with Sn by (2.2) and
+Map(P, Q) denotes the set of all maps from P to Q. The T -variety X(h) is what is called a GKM manifold
+and the image of the homomorphism in (2.5) is described in [20] as follows;
+(2.6)
+{f ∈ Map(Sn, Z[t1, . . . , tn]) | f(w) − f(w(i, j)) ∈ (tw(i) − tw(j)), for ∀w ∈ Sn, j < i ≤ h(j)},
+
+4
+M. MASUDA AND T. SATO
+where (i, j) denotes the transposition interchanging i and j. We note that the image of π∗(ti) ∈ π∗(H∗(BT )) ⊂
+H∗
+T (X(h)) by the homomorphism in (2.5) is the constant function ti ∈ Map(Sn, Z[t1, . . . , tn]).
+Guillemin and Zara [11] assigned a labeled graph to a GKM manifold. The labeled graph of X(h) is as
+follows. The vertex set is the fixed point set X(h)T = Sn. There is an edge between vertices w and v if and
+only if v = w(i, j) for some j ≤ i ≤ h(j), and the edge between w and w(i, j) is labeled by tw(i) − tw(j) up to
+sign.
+Example 2.1. Let n = 3. For h = (2, 3, 3) and h′ = (3, 3, 3), the labeled graphs of X(h) and X(h′) are
+drawn in Figure 2, where we use the one-line notation for each vertex.
+❞
+❞
+❞
+❞
+❞
+❞
+❍
+❍
+❍
+❍
+❍
+❍
+✟✟✟
+✟✟✟
+✟
+✟
+✟
+✟
+✟
+✟
+123
+321
+132
+312
+213
+231
+X(h)
+❞
+❞
+❞
+❞
+❞
+❞
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+❍
+✟✟✟
+✟✟✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+✟
+123
+321
+132
+312
+213
+231
+X(h′) = Fl(3)
+labels
+: t1 − t2
+: t2 − t3
+: t1 − t3
+Figure 2. The labeled graphs of X(h) and X(h′)
+In general, labeled graphs and their graph cohomologies are defined as follows.
+Definition 2.2. Let R be a ring. A labeled graph (Γ, α) consists of a graph Γ = (V, E) and a labeling
+α: E → R. The graph cohomology of a labeled graph (Γ, α) is defined as
+H∗(Γ, α) = {f ∈ Map(V, R) | f(w) − f(v) ∈ (α(e)) for ∀e = wv ∈ E}.
+The graph cohomology H∗(Γ, α) is a subring of Map(V, R) with the coordinate-wise sum and multiplication.
+Note that we may ignore the signs of the labels α(e) since (α(e)) = (−α(e)).
+The observation above shows that the graph cohomology of the labeled graph of X(h) coincides with
+H∗
+T (X(h)).
+Sending ti to tσ(i) for σ ∈ Sn and i ∈ [n] induces an action of Sn on Z[t1, . . . , tn]. Then, the module
+Map(Sn, Z[t1, . . . , tn]) becomes an Sn-module under what is called the dot action defined by
+(σ · f)(w) := σ(f(σ−1w)).
+As easily checked, the graph cohomology of X(h) is invariant under the dot action and H∗
+T (X(h)) becomes
+a module over Sn.
+Moreover, since the action of Sn preserves the ideal (π∗(t1), . . . , π∗(tn)), the action
+descends to H∗(X(h)) and H∗(X(h)) also becomes an module over Sn.
+Obviously, constant functions in Map(Sn, Z[t1, . . . , tn]) satisfy the condition in (2.6).
+They are ele-
+ments corresponding to π∗(H∗(BT )) ⊂ H∗
+T (X(h)).
+Below are three types of elements xi, yj,k, and τA
+in Map(Sn, Z[t1, . . . , tn]) which satisfy the condition in (2.6), so they are in H∗
+T (X(h)). Let
+⊥(h) : = {j ∈ [n − 1] | h(j − 1) = h(j) = j + 1}
+L(h) : = {j ∈ [n − 1] | h(j − 1) = j and h(j) = j + 1}
+(2.7)
+where we understand h(0) = 1.
+Definition 2.3.
+(1) For i ∈ [n], xi(w) := tw(i).
+(2) For j ∈ [n − 1] with j ∈ ⊥(h) and k ∈ [n],
+yj,k(w) :=
+�
+tk − tw(j+1)
+(if k ∈ {w(1), . . . , w(j)})
+0
+(otherwise).
+
+REGULAR SEMISIMPLE HESSENBERG VARIETIES
+5
+(3) For A ⊂ [n] with |A| ∈ L(h)
+τA(w) :=
+�
+tw(|A|) − tw(|A|+1)
+(if {w(1), . . . , w(|A|)} = A)
+0
+(otherwise).
+The cohomological degrees of the elements xk, yj,k, τA are two. One can easily check that the dot actions
+of σ ∈ Sn on these elements are given as follows:
+(2.8)
+σ · xk = xk,
+σ · yj,k = yj,σ(k),
+σ · τA = τσ(A).
+Remark 2.1. Here is a geometrical meaning of xk’s (regarded as elements in H2(X(h)) through the isomor-
+phism (2.4)). There is a nested sequence of tautological vector bundles over the flag variety Fl(n):
+F0 ⊂ F1 ⊂ F2 ⊂ · · · ⊂ Fn = Fl(n) × Cn
+where
+Fk := {(V•, v) ∈ Fl(n) × Cn | v ∈ Vk}
+and
+V• = ({0} = V0 ⊂ V1 ⊂ V2 ⊂ · · · ⊂ Vn = Cn).
+Then xk (k ∈ [n]) is the image of the first Chern class of the quotient line bundle Fk/Fk−1 over Fl(n) by the
+homomorphism
+ι∗ : H∗(Fl(n)) → H∗(X(h))
+induced from the inclusion map ι: X(h) → Fl(n). The dot action on H∗(Fl(n)) is trivial, so the image of ι∗
+must be contained in the ring of invariants H∗(X(h))Sn. In fact, it follows from [1, Theorems A and B] that
+the image of ι∗ agrees with H∗(X(h))Sn when tensoring with Q and
+(2.9)
+H∗(X(h))Sn ⊗ Q = Q[x1, . . . , xn]/(fh(1),1, . . . , fh(n),n)
+where
+(2.10)
+fh(j),j =
+j
+�
+k=1
+
+xk
+h(j)
+�
+ℓ=j+1
+(xk − xℓ)
+
+ .
+In particular, the Hilbert series of H∗(X(h))Sn is given by
+(2.11)
+Hilb(H∗(X(h))Sn, √q) =
+n−1
+�
+j=1
+[h(j) − j]q
+where the Hilbert series of a graded algebra A = �∞
+r=0 Ar over Z is defined as
+Hilb(A, q) :=
+∞
+�
+r=0
+(rankZAr)qr.
+Through the isomorphism (2.4), the elements xk, yj,k, τA determine elements in H2(X(h)), denoted by the
+same notation.
+Theorem 2.4 ([4, Theorem 5.1]). The elements
+{xk, yj,k, τA | k ∈ [n], j ∈ ⊥(h)\{n − 1}, A ⊂ [n] with |A| ∈ L(h)\{n − 1}}
+generate H2(X(h)) with relations
+(1) �n
+k=1 xk = 0,
+(2) �n
+k=1 yj,k = (x1 + · · · + xj) − jxj+1 for j ∈ ⊥(h)\{n − 1},
+(3) �
+|A|=j τA = xj − xj+1 for j ∈ L(h)\{n − 1}.
+Remark 2.2 (see Subsection 6.2 in [4] for more details). The element yj,k is defined by looking at the j-th
+column of the configuration associated to the Hessenberg function h. Similarly, one can define an element
+y∗
+i,k of H∗
+T (Hess(S, h)) by looking at the i-th row of the configuration as follows. For i ∈ [n], we define
+h∗(i) := min{j ∈ [n] | h(j) ≥ i},
+
+6
+M. MASUDA AND T. SATO
+so that the shaded boxes in the i-th row and under the diagonal in the configuration associated to h are at
+positions (i, ℓ) (h∗(i) ≤ ℓ < i). When h∗(i) = i − 1, we define
+(2.12)
+y∗
+i,k(w) :=
+�
+tk − tw(i−1)
+(k ∈ {w(i), . . . , w(n)})
+0
+(otherwise).
+One can see that y∗
+i.k is in H2
+T (Hess(S, h)) and we may replace yj,k’s for j ∈ ⊥(h)\{n − 1} in the generating
+set in Theorem 2.4 by y∗
+i,k’s for i ≥ 3 such that h∗(i) = h∗(i + 1) = i − 1.
+Example 2.2. When h = (4, 4, 4, 5, 6, 7, 11, 11, 11, 11) in Figure 1 (i.e. (a, b) = (3, 7)), we have
+⊥(h) = {3, 10},
+L(h) = {4, 5, 6},
+so Theorem 2.4 says that H2(X(h)) is generated by
+xk (k ∈ [11]),
+y3,k (k ∈ [11]),
+τA for A ⊂ [11] with |A| = 4, 5 or 6.
+Moreover, it follows from Remark 2.2 that y3,k above may be replaced by y∗
+8,k.
+3. Necessity
+In this section, we study a necessary condition on h for H∗(X(h)) to be generated in degree 2 as a ring.
+3.1. Moment maps. Let µ: Fl(n) → Rn be the standard moment map on the flag variety Fl(n). Its image is
+the permutohedron Πn obtained as the convex hull of the orbits of (1, 2, . . . , n) by permuting its coordinates.
+Indeed, if ew (w ∈ Sn) denotes the permutation flag associated with w, then we have
+µ(ew) = (w−1(1), . . . , w−1(n)) ∈ Rn
+(see [16, Lemma 3.1] for example). Let
+(3.1)
+Sr
+n := {w ∈ Sn | w(r) = n}.
+Then µ(Sr
+n) is the set of all vertices of Πn whose n-th coordinate is r. Therefore the projection
+πn : Πn → R,
+πn(x1, . . . , xn) = xn
+on the n-th coordinate takes minimum on S1
+n and maximum on Sn
+n. The composition of µ and πn
+(3.2)
+f := πn ◦ µ: Fl(n) → R
+is the moment map induced from the following S1-action on Cn
+(3.3)
+(z1, . . . , zn) → (z1, . . . , zn−1, gzn)
+(g ∈ S1 ⊂ C),
+and it is a Morse-Bott function.
+Let hj be the Hessenberg function obtained by removing all the boxes in the j-th row and all the boxes
+in the j-th column from its configuration (see Figure 3). To be precise, hj is given as follows.
+hj(i) =
+
+
+
+
+
+h(i)
+(i < j, h(i) < j)
+h(i) − 1
+(i < j, h(i) ≥ j)
+h(i + 1) − 1
+(i ≥ j)
+
+REGULAR SEMISIMPLE HESSENBERG VARIETIES
+7
+j-th row →
+↓
+j-th column
+h
+❀
+remove
+←
+տ
+↑
+❀
+hj
+Figure 3. The configuration corresponding to hj.
+The following is a key lemma in our argument.
+Lemma 3.1. The restriction maps
+H∗(X(h); Q) → H∗(X(h1); Q),
+H∗(X(h); Q) → H∗(X(hn); Q)
+are surjective.
+Proof. Let fh be the map f in (3.2) restricted to X(h), which is also a Morse-Bott function. The inverse
+image of the minimum value under fh is X(h1), so it follows from [19, Lemma 3.1] that the restriction map
+(3.4)
+H∗
+S1(X(h); Q) → H∗
+S1(X(h1); Q)
+is surjective, where the S1-action on X(h) is the induced one from the S1-action defined in (3.3). Since the
+S1-action on X(h1) is trivial, we have H∗
+S1(X(h1); Q) = H∗(BS1; Q)⊗ H∗(X(h1); Q) and hence the forgetful
+map H∗
+S1(X(h1); Q) → H∗(X(h1); Q) is surjective. Therefore, the surjectivity of (3.4) implies the surjectivity
+of the restriction map
+H∗(X(h); Q) → H∗(X(h1); Q)
+in ordinary cohomology. The same argument applied to −fh proves the statement for X(hn).
+✷
+Remark 3.1. The surjectivity of the above restriction maps (even with Z coefficients) can also be verified by
+GKM theory as follows. Recall that the inclusion of the fixed point set induces an injective homomorphism
+H∗
+T (X(h)) → H∗
+T (X(h)T ) ∼= Map(Sn, H∗(BT )). The equivariant cohomology H∗
+T (X(h)) has an H∗(BT )-
+module basis {σw,h | w ∈ Sn} (see [6, Definition 2.9 and Proposition 2.11]). It corresponds to a natural
+paving and then it is a ‘flow-up basis.’
+Note that any element of Sn
+n = Sn−1 is not greater than any
+element of Sn \ Sn
+n.
+The restriction of {σw,h | w ∈ Sn
+n} onto X(hn), that is, its restriction onto the
+fixed point set Sn
+n = X(hn)T as elements of Map(Sn, H∗(BT )), is a flow-up basis of H∗
+T (X(hn)). Hence
+H∗
+T (X(h)) → H∗
+T (X(hn)) is surjective, and then H∗(X(h)) → H∗(X(hn)) is also surjective. The surjectivity
+of H∗(X(h)) → H∗(X(h1)) can be verified by a similar argument.
+Given a Hessenberg function h, we obtain a smaller Hessenberg function by removing the first column and
+row or the last column and row repeatedly, i.e. by taking h1 or hn repeatedly. We call it a minor of h. The
+following corollary follows from Lemma 3.1.
+Corollary 3.2. Let h′ be a minor of h.
+If H∗(X(h); Q) is generated in degree 2 as a ring, then so is
+H∗(X(h′); Q).
+An easy argument shows that h being of the form (1.1) can be rephrased as follows.
+Proposition 3.3. The Hessenberg function h is of the form (1.1) if and only if h has neither
+(α, β, . . . , β), (β − 1, . . . , β − 1, β, . . . , β
+�
+��
+�
+α
+) for 3 ≤ α < β, nor (2, γ − 1, . . . , γ − 1, γ, γ) for γ ≥ 5
+as its minor.
+
+8
+M. MASUDA AND T. SATO
+Recall that if h† denotes the Hessenberg function obtained by flipping the configuration of h along the
+anti-diagonal, then X(h†) ∼= X(h) as varieties. Therefore
+X((α, β, . . . , β)) ∼= X((β − 1, . . . , β − 1, β, . . . , β
+�
+��
+�
+α
+)).
+Here, we know that H∗(X((α, β, . . . , β)); Q) is not generated in degree 2 for 3 ≤ α < β by [2, Theorem 4.3].
+Thus, it suffices to treat the last case in Proposition 3.3, which we shall discuss in the next subsection.
+3.2. The case h = (2, n − 1, . . . , n − 1, n, n). In this subsection we prove the following proposition.
+Proposition 3.4. H∗(X(h); Q) is not generated in degree 2 when h = (2, n − 1, . . . , n − 1, n, n) for n ≥ 5.
+Some computation is involved in the proof of this proposition but the idea of the proof is simple. We
+compute the Poincar´e polynomial of X(h) using Theorem 2.1(4). On the other hand, using explicit generators
+of H2(X(h)) by [4], we compute an upper bound of the Hilbert series of the subring of H∗(X(h)) generated
+by H2(X(h)). Then it turns out that the latter is strictly smaller than the former at a certain degree.
+3.2.1. Poincar´e polynomial of X(h). The following proposition, which easily follows from Theorem 2.1(4),
+enables us to compute the Poincar´e polynomial of X(h) inductively.
+Proposition 3.5 ([4, Proposition 3.1]).
+(3.5)
+Poin(X(h), √q) =
+n
+�
+j=1
+qh(j)−j Poin(X(hj), √q).
+Using the proposition above, the Poincar´e polynomial of X(h) is explicitly computed as follows when
+h = (h(1), n, . . . , n).
+Proposition 3.6 ([2]). When h = (h(1), n . . . , n), we have
+(3.6)
+Poin(X(h), √q) = [h(1)]q[n − 1]q! + (n − 1)qh(1)−1[n − h(1)]q[n − 2]q!,
+where
+[m]q = 1 − qm
+1 − q ,
+[m]q! = [1]q[2]q · · · [m]q =
+m
+�
+j=1
+1 − qj
+1 − q .
+Now, let h = (2, n − 1, . . . , n − 1, n, n) and set
+Pn(q) := Poin(X(h), √q).
+Lemma 3.7. For n ≥ 5, the following recurrence formula holds
+Pn(q) = (1 + q)2[n − 2]q! + (n − 2)(q + q2)[n − 3]q[n − 3]q!
++ (n − 1)(q + qn−3) {(1 + q)[n − 3]q! + (n − 3)q[n − 4]q[n − 4]q!}
++ (q + q2 + · · · + qn−4)Pn−1(q).
+Proof. Let Fn(q) denote the right-hand side of (3.6) with h(1) = 2, that is,
+(3.7)
+Fn(q) := (1 + q)[n − 1]q! + (n − 1)q[n − 2]q[n − 2]q!.
+Then we have
+Poin(X(h1), √q) = Poin(X(hn), √q) = Fn−1(q)
+Poin(X(h2), √q) = Poin(X(hn−1), √q) = (n − 1)Fn−2(q)
+Poin(X(hj), √q) = Pn−1(q)
+(3 ≤ j ≤ n − 2),
+
+REGULAR SEMISIMPLE HESSENBERG VARIETIES
+9
+where we note that X(h2) consists of n − 1 copies of Fl(n − 2). Hence, by (3.5), we have
+Pn(q) = qFn−1(q) + (n − 1)qn−3Fn−2(q) + (qn−4 + · · · + q)Pn−1(q)
++ (n − 1)qFn−2(q) + Fn−1(q)
+= (1 + q)Fn−1(q) + (n − 1)(q + qn−3)Fn−2(q) + (q + · · · + qn−4)Pn−1(q).
+Combining this equation with (3.7), we obtain the desired equation.
+✷
+Lemma 3.8. For n ≥ 4, let
+Qn(q) = (1 + 2nq + n(n − 1)q2)[n − 2]q! + n(n − 3)
+2
+qn−3.
+Then we have
+Pn(q) ≡ Qn(q)
+mod (qn−2).
+In other words, Pn(q) and Qn(q) coincide up to degree n − 3.
+Proof. We prove the lemma by induction on n. When n = 4, we have
+P4(q) = 1 + 11q + 11q2 + q3,
+Q4(q) = 1 + 11q + 20q2 + 12q3,
+and the lemma is true for n = 4.
+Let n be given and suppose that the lemma is true for n − 1, that is,
+(3.8)
+Pn−1(q) ≡ Qn−1(q)
+mod (qn−3).
+Hereafter, in this proof, all congruences will be taken modulo qn−2 unless otherwise stated. Since we have
+(q + q2)[n − 3]q[n − 3]q! ≡ (q + q2)[n − 2]q!
+q2[n − 4]q[n − 4]q! ≡ q2[n − 3]q!,
+the recurrence formula in Lemma 3.7 reduces to the following congruence relation:
+Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q! + (n − 1)(q + (n − 2)q2)[n − 3]q!
++ (n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q).
+(3.9)
+It follows from (3.8) and the definition of Qn that the sum of the last two terms above becomes as follows.
+(n − 1)qn−3 + (q + · · · + qn−4)Pn−1(q)
+≡ (n − 1)qn−3 +
+�
+1 + (2n − 2)q + (n − 1)(n − 2)q2�
+(q + · · · + qn−4)[n − 3]q! + (n − 1)(n − 4)
+2
+qn−3
+=
+�
+1 − q + (n − 1)q(1 − q) + nq + (n − 1)2q2�
+(q + · · · + qn−4)[n − 3]q! + (n − 1)(n − 2)
+2
+qn−3
+≡
+�
+q − qn−3 + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4)
+�
+[n − 3]q! + (n − 1)(n − 2)
+2
+qn−3
+≡
+�
+q + (n − 1)q2 + (nq + (n − 1)2q2)(q + · · · + qn−4)
+�
+[n − 3]q! + n(n − 3)
+2
+qn−3
+By substituting it to (3.9), we obtain
+Pn(q) ≡ (1 + nq + (n − 1)q2)[n − 2]q!
++
+�
+(nq + (n − 1)2q2) + (nq + (n − 1)2q2)(q + · · · + qn−4)
+�
+[n − 3]q! + n(n − 3)
+2
+qn−3
+≡ (1 + nq + (n − 1)q2)[n − 2]q! + (nq + (n − 1)2q2)[n − 2]q! + n(n − 3)
+2
+qn−3
+= (1 + 2nq + n(n − 1)q2)[n − 2]q! + n(n − 3)
+2
+qn−3
+= Qn(q).
+
+10
+M. MASUDA AND T. SATO
+This completes the induction step and the lemma has been proved.
+✷
+3.2.2. Hilbert series of the subring generated by H2(X(h)). When h = (2, n − 1, . . ., n − 1, n, n) for n ≥ 5, we
+first observe H2(X(h)). By (2.7), we have
+⊥(h) = {n − 2},
+L(h) = {1, n − 1}.
+Therefore, it follows from Theorem 2.4 that H2(X(h)) is generated by the following elements
+(3.10)
+xk,
+yk := yn−2,k,
+τk := τ{k}
+(k ∈ [n]),
+where
+xk(w) = tw(k),
+yk(w) = yn−2,k(w) =
+�
+tk − tw(n−1)
+(if k ∈ {w(1), . . . , w(n − 2)})
+0
+(otherwise),
+τk(w) = τ{k}(w) =
+�
+tw(1) − tw(2)
+(if k = w(1))
+0
+(otherwise)
+(3.11)
+for w ∈ Sn by Definition 2.3, and
+(3.12)
+n
+�
+k=1
+yk = x1 + · · · + xn−2 − (n − 2)xn−1,
+n
+�
+k=1
+τk = x1 − x2
+by Theorem 2.4. We also have
+σ · xk = xk,
+σ · yk = yσ(k),
+σ · τk = τσ(k)
+for σ ∈ Sn by (2.8).
+To make the following argument clearer, we introduce elements ρk for k ∈ [n] defined by
+(3.13)
+ρk(w) :=
+�
+tw(n−1) − tw(n)
+(if k = w(n))
+0
+(otherwise).
+Similarly to τk, the ρk satisfies the condition (2.6) so that it defines an element of H2
+T (X(h)) and H2(X(h))
+and
+(3.14)
+n
+�
+k=1
+ρk = xn−1 − xn,
+σ · ρk = ρσ(k)
+for σ ∈ Sn.
+An elementary check shows that
+(yk − yℓ)(w) − (ρk − ρℓ)(w) = tk − tℓ
+(k, ℓ ∈ [n], w ∈ Sn)
+and hence yk − yℓ = ρk − ρℓ in H2(X(h)). Moreover, �n
+k=1 yk and �n
+k=1 ρk are both linear polynomials in
+xi’s by (3.12) and (3.14), so we may replace yk’s in the generating set (3.10) by ρk’s. Namely H2(X(h)) is
+generated by
+xk,
+τk,
+ρk
+(k ∈ [n])
+with relations
+(3.15)
+n
+�
+k=1
+xk = 0,
+n
+�
+k=1
+τk = x1 − x2,
+n
+�
+k=1
+ρk = xn−1 − xn,
+and the actions of σ ∈ Sn on those generators are given by
+(3.16)
+σ · xk = xk,
+σ · τk = τσ(k),
+σ · ρk = ρσ(k).
+
+REGULAR SEMISIMPLE HESSENBERG VARIETIES
+11
+Our purpose is to find a sharp upper bound of the Hilbert series of the subring R(h) of H∗(X(h)) generated
+by H2(X(h)). Let A(h) be the subring of H∗(X(h)) generated by xk’s and we regard R(h) as a module over
+A(h). It follows from (3.11) and (3.13) that
+τkτℓ =
+�
+(x1 − x2)τk
+(k = ℓ)
+0
+(k ̸= ℓ),
+ρkρℓ =
+�
+(xn−1 − xn)ρk
+(k = ℓ)
+0
+(k ̸= ℓ),
+τkρk = 0.
+Therefore, R(h) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h). The subring
+A(h) itself is a submodule of R(h) over A(h). We consider three other submodules of R(h) over A(h):
+B(h) :={
+n
+�
+k=1
+bkτk | bk ∈ A(h),
+n
+�
+k=1
+bk = 0},
+C(h) :={
+n
+�
+k=1
+ckρk | ck ∈ A(h),
+n
+�
+k=1
+ck = 0},
+D(h) :={
+�
+1≤i,j≤n
+dijτiρj | dij ∈ A(h),
+n
+�
+j=1
+dij = 0 for i ∈ [n],
+n
+�
+i=1
+dij = 0 for j ∈ [n]}
+(3.17)
+where dkk = 0 for k ∈ [n]. Note that A(h)⊗Q agrees with the ring of invariants H∗(X(h); Q)Sn as mentioned
+in Remark 2.1.
+Lemma 3.9. R(h) is additively generated by A(h), B(h), C(h), and D(h) when tensoring with Q.
+Proof. Since H∗(X(h)) is generated by 1, τk, ρk (k ∈ [n]), and τiρj (i ̸= j ∈ [n]) as a module over A(h), it
+suffices to show that any element of the form
+(3.18)
+n
+�
+k=1
+bkτk +
+n
+�
+k=1
+ckρk +
+�
+1≤i,j≤n
+dijτiρj
+(bk, ck, dij ∈ A(h), dkk = 0)
+can be expressed as a sum of elements in A(h), B(h), C(h), and D(h) when tensoring with Q.
+Step 1. Set b := �n
+k=1 bk and c := �n
+k=1 ck. Since �n
+k=1 τk = x1 −x2 and �n
+k=1 ρk = xn−1 −xn by (3.15),
+we have
+n
+�
+k=1
+bkτk +
+n
+�
+k=1
+ckρk =
+n
+�
+k=1
+�
+bk − b
+n
+�
+τk + b
+n(x1 − x2) +
+n
+�
+k=1
+�
+ck − c
+n
+�
+ρk + c
+n(xn−1 − xn).
+Here the two sums at the right hand side above respectively belong to B(h) ⊗ Q and C(h) ⊗ Q, and the
+remaining two terms belong to A(h) ⊗ Q.
+Step 2. As for the last term in (3.18), since �n
+i=1 τi = x1 − x2, we have
+�
+1≤i,j≤n
+dijτiρj =
+n
+�
+j=1
+� n
+�
+i=1
+�
+dij − dj
+n
+�
+τi
+�
+ρj +
+n
+�
+j=1
+dj
+n (x1 − x2)ρj
+=
+�
+1≤i,j≤n
+˜dijτiρj +
+n
+�
+j=1
+dj
+n (x1 − x2)ρj
+(3.19)
+where
+dj :=
+n
+�
+i=1
+dij
+and
+˜dij := dij − dj
+n .
+The last sum in (3.19) is a sum of elements in A(h) ⊗ Q and C(h) ⊗ Q by Step 1. We shall show that the
+sum �
+1≤i,j≤n ˜dijτiρj in (3.19) is a sum of elements in A(h) ⊗ Q, B(h) ⊗ Q, and D(h) ⊗ Q. We note that
+(3.20)
+n
+�
+i=1
+˜dij =
+n
+�
+i=1
+�
+dij − dj
+n
+�
+=
+n
+�
+i=1
+dij − dj = 0
+
+12
+M. MASUDA AND T. SATO
+and set
+(3.21)
+˜di :=
+n
+�
+j=1
+˜dij.
+Since �n
+j=1 ρj = xn−1 − xn, we have
+(3.22)
+�
+1≤i,j≤n
+˜dijτiρj =
+n
+�
+i=1
+
+
+n
+�
+j=1
+�
+˜dij −
+˜di
+n
+�
+ρj
+
+ τi +
+n
+�
+i=1
+˜di
+n (xn−1 − xn)τi.
+Here the second sum at the right hand side of (3.22) is a sum of elements in A(h) ⊗ Q and B(h) ⊗ Q by Step
+1. As for the coefficients ˜dij −
+˜di
+n of τiρj in the first sum at the right hand side of (3.22), it follows from
+(3.20) and (3.21) that we have
+n
+�
+i=1
+�
+˜dij −
+˜di
+n
+�
+=
+n
+�
+i=1
+˜dij − 1
+n
+n
+�
+i=1
+˜di = − 1
+n
+n
+�
+i=1
+n
+�
+j=1
+˜dij = −
+n
+�
+j=1
+� n
+�
+i=1
+˜dij
+�
+= 0,
+n
+�
+j=1
+�
+˜dij −
+˜di
+n
+�
+=
+n
+�
+j=1
+˜dij − ˜di = 0.
+Thus, the first sum at the right hand side of (3.22) belongs to D(h) ⊗ Q. This completes the proof of the
+lemma.
+✷
+We shall calculate upper bounds of the Hilbert series of A(h), B(h), C(h), and D(h).
+Hilbert series of A(h). Since A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and h = (2, n − 1, . . . , n − 1, n, n) in our case,
+it follows from (2.11) that
+(3.23)
+Hilb(A(h), √q) =
+n−1
+�
+j=1
+[h(j) − j]q = (1 + q)2[n − 2]q!.
+Hilbert series of B(h). It follows from(3.11) that (x1 − tk)τk vanishes at every w ∈ Sn, so we have
+(3.24)
+(x1 − tk)τk = 0
+in H∗
+T (X(h))
+and hence
+x1τk = 0
+in H∗(X(h)).
+Therefore, B(h) is indeed a module over A(h)/(x1). Here
+A(h)/(x1) ⊗ Q = A(h1) ⊗ Q
+by (2.9) and (2.10). Since h1 = (n − 2, . . . , n − 2, n − 1, n − 1), it follows from (2.11) that
+Hilb(A(h)/(x1), √q) =
+n−2
+�
+j=1
+[h1(j) − j]q = (1 + q)[n − 2]q!.
+Since B(h) is a module over A(h)/(x1) generated by τi − τi+1 (i ∈ [n − 1]) and the cohomological degrees of
+τk’s are two, we obtain an upper bound of Hilb(B(h), q) as follows:
+(3.25)
+Hilb(B(h), √q) ≤ (n − 1)q Hilb(A(h)/(x1), √q) = (n − 1)(q + q2)[n − 2]q!.
+Here �∞
+i=0 aiqi ≤ �∞
+i=0 biqi (ai, bi ∈ Z) means that ai ≤ bi for all i’s.
+Hilbert series of C(h). To f ∈ Map(Sn, Z[t1, . . . , tn]) we associate f ∨ ∈ Map(Sn, Z[t1, . . . , tn]) defined by
+f ∨(w) := f(ww0)
+for w ∈ Sn,
+where w0 denotes the longest element in Sn, i.e. w0 = n n − 1 · · · 2 1 in one-line notation. This defines an
+involution on Map(Sn, Z[t1, . . . , tn]) and one can easily check that
+x∨
+k = xn−k+1,
+τ ∨
+k = −ρk,
+ρ∨
+k = −τk
+
+REGULAR SEMISIMPLE HESSENBERG VARIETIES
+13
+from (3.11) and (3.13). Hence the involution gives an isomorphism between B(h) and C(h), and the same
+inequality as (3.25) holds for C(h), i.e.
+(3.26)
+Hilb(C(h), √q) ≤ (n − 1)(q + q2)[n − 2]q!.
+Hilbert series of D(h). We have x1τk = 0 by (3.24). Similarly we have xnρk = 0 since (x1τk)∨ = −xnρk.
+(The fact xnρk = 0 also follows from the definition (3.11) and (3.13) of xk and ρk.) Therefore, D(h) is indeed
+a module over A(h)/(x1, xn).
+As mentioned in Remark 2.1, A(h) ⊗ Q = H∗(X(h))Sn ⊗ Q and it is the image of the restriction map
+ι∗ : H∗(Fl(n)) → H∗(X(h)). Therefore, A(h)/(x1, xn) is the image of the restriction map from H∗(Fl(n−2))
+and hence
+Hilb(A(h)/(x1, xn), √q) ≤ [n − 2]q!.
+(In fact, the equality holds above.) There are 2n relations among dij (i ̸= j) in the definition (3.17) of D(h),
+but one relation can be obtained from the other 2n−1 relations because �n
+i=1
+��n
+j=1 dij
+�
+= �n
+j=1 (�n
+i=1 dij).
+Moreover, there are n(n − 1) number of dij’s and the cohomological degree of τiρj is four. Thus
+(3.27)
+Hilb(D(h), √q) ≤ Hilb(A(h)/(x1, xn), √q) {n(n − 1) − (2n − 1)} q2 ≤ (n2 − 3n + 1)q2[n − 2]q!.
+Proof of Proposition 3.4. It follows from Lemma 3.9, (3.23), (3.25), (3.26), and (3.27) that
+Hilb(R(h), √q) ≤ (1 + q)2[n − 2]q! + 2(n − 1)(q + q2)[n − 2]q! + (n2 − 3n + 1)q2[n − 2]q!
+= (1 + 2nq + n(n − 1)q2)[n − 2]q!.
+The coefficient of qn−3 in the last term above is less than that of Pn(q) in Lemma 3.8 by n(n − 3)/2, proving
+the proposition.
+✷
+4. Sufficiency
+The purpose of this section is to prove the following proposition, which implies the sufficiency of Theorem
+1.1.
+Proposition 4.1. When h is of the form (1.1), the equivariant cohomology H∗
+T (X(h)) is generated in degree
+2 as an algebra over H∗(BT ).
+By Theorem 2.1(3), X(h) is not connected when h(k) = k for some 1 ≤ k ≤ n − 1. In this case, a flag
+V• = (V0 ⊂ V1 ⊂ · · · ⊂ Vn) ∈ X(h) is of the form Vk = ⟨ei1, ei2, . . . , eik⟩ for some {i1, . . . , ik} ⊂ [n], where ei
+is the i-th standard basis vector of Cn. Therefore, decomposing V• into two flags (V0 ⊂ V1 ⊂ · · · ⊂ Vk) and
+(V ′
+0 ⊂ V ′
+1 ⊂ · · · ⊂ V ′
+n−k), where V ′
+i = Vk+i/Vk, one can see that X(h) is the disjoint union of
+�n
+k
+�
+copies of
+X(h1) × X(h2), where h1 and h2 are the Hessenberg function obtained by restricting h onto intervals [k] and
+[k + 1, n], respectively. Each copy corresponds to the choice of a k-subset {i1, . . . , ik} ⊂ [n]. To be precise,
+h2 : [n − k] → [n − k] is given by shift−1
+k
+◦ h ◦ shiftk, where shiftk : [n − k] → [k + 1, n] shifts integers by k.
+Suppose h is of the form (1.1) and 1 ≤ r ≤ n. Then
+X(hr) is not connected ⇐⇒ a + 1 ≤ r ≤ b
+by Theorem 2.1(3) and that hr is also of the form (1.1) when r < a + 1 or r > b. When a + 1 ≤ r ≤ b, each
+connected component of X(hr) is isomorphic to X(h1) × X(h2) and both h1 and h2 are of the form (1.1).
+Let Γ(Sn, h) denote the labeled graph of X(h). Recall that H∗
+T (X(h)) ∼= H∗(Γ(Sn, h)). For the subset
+Sr
+n ⊂ Sn in (3.1), let Γ(Sr
+n, h) be the induced labeled subgraph of Γ(Sn, h) on the subset Sr
+n of vertices,
+and let Γ0(Sr
+n, h) denote a connected component of Γ(Sr
+n, h).
+Lemma 4.2. When h is of the form (1.1), the restriction map H2(Γ(Sn, h)) → H2(Γ0(Sr
+n, h)) is surjective.
+We admit the lemma and complete the proof of Proposition 4.1.
+Before that, we shall observe that
+Γ0(Sr
+n, h) is essentially a connected component of a labeled graph of X(hr). Indeed, for 1 ≤ r ≤ n, let cr be
+the cyclic permutation (r r + 1 r + 2 · · · n) and
+ϕr : Γ0(Sr
+n, h) → Γ0(Sn−1, hr)
+
+14
+M. MASUDA AND T. SATO
+a graph isomorphism defined by ϕr(w) = wcr for w ∈ Sr
+n. When i, j ̸= r, the (i, j)-th box in the configuration
+for h corresponds to the (c−1
+r (i), c−1
+r (j))-th box in the configuration for hr (see Figure 3). In particular,
+v = w(i, j) corresponds to vcr = wcr(c−1
+r (i), c−1
+r (j)) and the edges between these vertices have the same
+label tw(i) − tw(j). Therefore, ϕr induces an isomorphism
+(4.1)
+ϕ∗
+r : H∗(Γ0(Sn−1, hr))
+∼
+=
+−→ H∗(Γ0(Sr
+n, h))
+of graded algebras over H∗(BT ).
+Proof of Proposition 4.1. Recall that H∗
+T (X(h)) ∼= H∗(Γ(Sn, h)). We prove the proposition by induction on
+n. Let 1 ≤ r ≤ n. For any z ∈ H∗(Γ(Sn, h)) that vanishes on �r−1
+j=1 Sj
+n, it is sufficient to show the existence
+of a polynomial f in elements of H2(Γ(Sn, h)) such that z − f vanishes on �r
+j=1 Sj
+n. Then the induction on
+r proves the proposition. We shall show the existence of f by division into cases according to the value of r.
+Case 1. The case 1 ≤ r ≤ a. In this case, Γ(Sr
+n, h) is connected. We note that z vanishes on �r−1
+j=1 Sj
+n
+and this implies that z(w) for w ∈ Sr
+n decomposes as follows:
+(4.2)
+z(w) =
+
+
+r−1
+�
+j=1
+(tw(j) − tn)
+
+ g(w),
+g ∈ H∗(Γ(Sr
+n, h)).
+Indeed, for w ∈ Sr
+n, we have w(r) = n and w(j, r) ∈ Sj
+n. If j ≤ r − 1, then there is an edge in the graph
+Γ(Sn, h) between the vertices w and w(j, r). The label on the edge is tw(j) −tw(r) = tw(j) −tn and z vanishes
+at w(j, r) ∈ Sj
+n (j ≤ r − 1) by assumption. Therefore z(w) is divisible by the product in the big parenthesis
+in (4.2) and g ∈ Map(Sr
+n, H∗(BT )). Furthermore, one can easily check that the g is indeed in H∗(Γ(Sr
+n, h))
+since z is in H∗(Γ(Sn, h)).
+Since H∗(Γ(Sr
+n, h)) ∼= H∗(Γ(Sn−1, hr)) by (4.1), g is a polynomial in elements of H2(Γ(Sr
+n, h)) by induc-
+tion on n. Moreover, by Lemma 4.2, there is a polynomial ˜g in H2(Γ(Sn, h)) which coincides with g on Sr
+n.
+On the other hand, �r−1
+j=1(xj −tn) coincides with the product in (4.2) on Sr
+n since xj(w) = tw(j) by definition
+of xj, and vanishes on �r−1
+j=1 Sj
+n since xj(w) = tw(j) = tn for w ∈ Sj
+n. Therefore,
+� �r−1
+j=1(xj − tn)
+�
+˜g coincides
+with the element z on �r
+j=1 Sj
+n. Thus
+� �r−1
+j=1(xj − tn)
+�
+˜g is a desired polynomial f.
+Case 2. The case r = a + 1. Similarly to Case 1, z(w) for w ∈ Sa+1
+n
+decomposes as follows:
+(4.3)
+z(w) =
+
+
+a
+�
+j=1
+(tw(j) − tn)
+
+ g(w),
+g ∈ H∗(Γ(Sa+1
+n
+, h)).
+Note that Γ(Sa+1
+n
+, h) is not connected. Two vertices v, w ∈ Sa+1
+n
+lie in the same connected component if
+and only if
+{v(1), . . . , v(a)} = {w(1), . . . , w(a)} ⊂ [n − 1].
+For K := {k1, . . . , ka} ⊂ [n − 1], we consider the element ρK defined by
+ρK =
+a
+�
+j=1
+ya,kj,
+where
+ya,k(w) =
+�
+tk − tw(a+1)
+(k ∈ {w(1), . . . , w(a)})
+0
+(k /∈ {w(1), . . . , w(a)})
+by definition. Therefore, since w(a + 1) = n for w ∈ Sa+1
+n
+, we have
+ρK(w) =
+��a
+j=1(tw(j) − tn)
+(K = {w(1), . . . , w(a)})
+0
+(K ̸= {w(1), . . . , w(a)}).
+Hence ρK coincides with the product in the big parentheses of (4.3) on the connected component
+(4.4)
+{w ∈ Sa+1
+n
+| w([a]) = K}
+
+REGULAR SEMISIMPLE HESSENBERG VARIETIES
+15
+and vanishes on the other components. Since n /∈ K and w(j) = n for w ∈ Sj
+n, ρK also vanishes on �a
+j=1 Sj
+n.
+On the other hand, the element g in (4.3) restricted to the connected component (4.4) is obtained as the
+restriction of a polynomial ˜gK in H2(Γ(Sn, h)) similarly to Case 1. Therefore, we obtain a desired polynomial
+f as
+�
+K⊂[n−1], |K|=a
+ρK˜gK.
+Case 3. The case a + 2 ≤ r ≤ b. In this case, z(w) for w ∈ Sr
+n decomposes as follows:
+(4.5)
+z(w) = (tw(r−1) − tw(r))g(w),
+g ∈ H∗(Γ(Sr
+n, h)).
+Similarly to Case 2, Γ(Sr
+n, h) is not connected and two vertices v, w ∈ Sr
+n lie in the same connected component
+if and only if
+{v(1), . . . , v(r − 1)} = {w(1), . . . , w(r − 1)} ⊂ [n − 1].
+For A ⊂ [n − 1] with |A| = r − 1, we have
+τA(w) =
+�
+tw(r−1) − tw(r)
+(A = {w(1), . . . , w(r − 1)})
+0
+(A ̸= {w(1), . . . , w(r − 1)})
+by definition. Hence, τA coincides with the factor of the right-hand side of (4.5) on the connected component
+{w ∈ Sr
+n | w([r − 1]) = A}, and vanishes on the other connected components. Since n /∈ A and w(j) = n for
+w ∈ Sj
+n, τA also vanishes on �r−1
+j=1 Sj
+n. Therefore, similarly to Case 2, we obtain a desired polynomial f as
+�
+A⊂[n−1], |A|=r−1
+τA˜gA, where ˜gA is a polynomial in H2(Γ(Sn, h)).
+Case 4. The case b + 1 ≤ r ≤ n. In this case, z(w) for w ∈ Sr
+n decomposes as follows:
+(4.6)
+z(w) =
+
+
+r−1
+�
+j=b
+(tn − tw(j))
+
+ g(w),
+g ∈ H∗(Γ(Sr
+n, h)).
+Similarly to Case 1, X(hr) is connected and g is the restriction of a polynomial ˜g in H2(Γ(Sn, h)).
+We consider the element y∗
+b+1,n ∈ H∗(Γ(Sn, h)) in Remark 2.2, which is defined as
+y∗
+b+1,n(w) =
+�
+tn − tw(b)
+(n ∈ {w(b + 1), . . . , w(n)})
+0
+(n /∈ {w(b + 1), . . . , w(n)}).
+Then
+
+y∗
+b+1,n
+r−1
+�
+j=b+1
+(tn − xj)
+
+ (w) =
+��r−1
+j=b(tn − tw(j))
+(n ∈ {w(b + 1), . . . , w(n)})
+0
+(n /∈ {w(b + 1), . . . , w(n)}).
+Hence y∗
+b+1,n
+�r−1
+j=b+1(tn − xj) coincides with the product in the big parentheses of (4.6) on Sr
+n, and vanishes
+on �r−1
+j=1 Sj
+n. Therefore,
+�
+y∗
+b+1,n
+�r−1
+j=b+1(tn − xj)
+�
+˜g is a desired polynomial f.
+✷
+Finally we give a proof of Lemma 4.2.
+Proof of Lemma 4.2. It follows from Theorem 2.4 and Remark 2.2 that when h is of the form (1.1), the
+elements in
+{xi, ya,k, τA, ti | i, k ∈ [n], A ⊂ [n], a + 1 ≤ |A| < b}
+span H2(Γ(Sn, h)). Through the isomorphism (4.1), one can find generators of H2(Γ0(Sr
+n, h)) which corre-
+spond to the generators of H2(Γ0(Sn−1, hr)). They are given as restrictions of
+xi for i ∈ [n], i ̸= r,
+ti for i ∈ [n],
+and the following elements in H2(Γ(Sn, h)).
+Case 1. When 1 ≤ r ≤ a,
+ya,k for k ∈ [n − 1],
+τA⊔{n} for A ⊂ [n − 1], a ≤ |A| < b − 1.
+
+16
+M. MASUDA AND T. SATO
+Case 2. When r = a + 1, for a connected component Γ0(Sa+1
+n
+, h) which contains σ ∈ Sa+1
+n
+;
+τB⊔σ([a+1]) for B ⊂ σ([n]\[a + 1]), 1 ≤ |B| < b − (a + 1).
+Case 3. When a + 1 < r ≤ b, for a connected component Γ0(Sr
+n, h) which contains σ ∈ Sr
+n;
+ya,k for k ∈ [n − 1],
+τA for A ⊂ σ([r − 1]), a + 1 ≤ |A| < r − 1,
+τB⊔σ([r]) for B ⊂ σ([n] \ [r]), 1 ≤ |B| < b − r.
+Case 4. When b < r ≤ n,
+ya,k for k ∈ [n − 1],
+τA for A ⊂ [n − 1], a + 1 ≤ |A| < b.
+This proves the lemma.
+✷
+Acknowledgment.
+We thank Yunhyung Cho for his help on moment map. Masuda was supported in part by JSPS Grant-
+in-Aid for Scientific Research 22K03292 and a HSE University Basic Research Program. This work was
+partly supported by Osaka Central Advanced Mathematical Institute (MEXT Joint Usage/Research Center
+on Mathematics and Theoretical Physics JPMXP0619217849).
+References
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+Osaka Metropolitan University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan.
+Email address: [email protected]
+Osaka Metropolitan University Advanced Mathematical Institute, Sumiyoshi-ku, Osaka 558-8585, Japan.
+Email address: [email protected]
+

99E2T4oBgHgl3EQfQQZv/content/tmp_files/2301.03768v1.pdf.txt

@@ -0,0 +1,1247 @@
+Prepared for submission to JHEP
+Search for lepton-flavor-violating τ decays into a
+lepton and a vector meson using the full Belle data
+sample
+The Belle Collaboration
+N. Tsuzuki
+, K. Inami
+, I. Adachi
+, H. Aihara
+, D. M. Asner
+, H. Atmacan
+,
+T. Aushev
+, R. Ayad
+, V. Babu
+, Sw. Banerjee
+, P. Behera
+, K. Belous
+,
+J. Bennett
+, M. Bessner
+, B. Bhuyan
+, T. Bilka
+, D. Biswas
+, D. Bodrov
+,
+J. Borah
+, A. Bozek
+, M. Braˇcko
+, P. Branchini
+, T. E. Browder
+, A. Budano
+,
+M. Campajola
+, D. ˇCervenkov
+, M.-C. Chang
+, B. G. Cheon
+, K. Chilikin
+,
+H. E. Cho
+, K. Cho
+, S.-J. Cho
+, S.-K. Choi
+, Y. Choi
+, S. Choudhury
+,
+D. Cinabro
+, J. Cochran
+, S. Das
+, N. Dash
+, G. De Nardo
+, G. De Pietro
+,
+R. Dhamija
+, F. Di Capua
+, Z. Doleˇzal
+, T. V. Dong
+, D. Dossett
+, S. Dubey
+,
+D. Epifanov
+, T. Ferber
+, D. Ferlewicz
+, B. G. Fulsom
+, V. Gaur
+, A. Giri
+,
+P. Goldenzweig
+, Y. Guan
+, K. Gudkova
+, X. Han
+, T. Hara
+, K. Hayasaka
+,
+H. Hayashii
+, M. T. Hedges
+, D. Herrmann
+, W.-S. Hou
+, C.-L. Hsu
+,
+T. Iijima
+, N. Ipsita
+, A. Ishikawa
+, R. Itoh
+, M. Iwasaki
+, W. W. Jacobs
+,
+E.-J. Jang
+, S. Jia
+, Y. Jin
+, T. Kawasaki
+, C. Kiesling
+, C. H. Kim
+,
+D. Y. Kim
+, K.-H. Kim
+, Y.-K. Kim
+, K. Kinoshita
+, P. Kodyˇs
+, T. Konno
+,
+A. Korobov
+, S. Korpar
+, E. Kovalenko
+, P. Kriˇzan
+, P. Krokovny
+, M. Kumar
+,
+K. Kumara
+, A. Kuzmin
+, Y.-J. Kwon
+, S. C. Lee
+, J. Li
+, L. K. Li
+, Y. Li
+,
+J. Libby
+, K. Lieret
+, Y.-R. Lin
+, D. Liventsev
+, Y. Ma
+, A. Martini
+,
+M. Masuda
+, K. Matsuoka
+, D. Matvienko
+, S. K. Maurya
+, F. Meier
+,
+M. Merola
+, K. Miyabayashi
+, R. Mizuk
+, G. B. Mohanty
+, M. Nakao
+,
+Z. Natkaniec
+, A. Natochii
+, L. Nayak
+, M. Niiyama
+, N. K. Nisar
+, S. Nishida
+,
+S. Ogawa
+, H. Ono
+, P. Oskin
+, G. Pakhlova
+, T. Pang
+, S. Pardi
+, H. Park
+,
+J. Park
+, S.-H. Park
+, A. Passeri
+, S. Paul
+, T. K. Pedlar
+, R. Pestotnik
+,
+L. E. Piilonen
+, T. Podobnik
+, E. Prencipe
+, M. T. Prim
+, A. Rostomyan
+,
+N. Rout
+, G. Russo
+, S. Sandilya
+, A. Sangal
+, L. Santelj
+, V. Savinov
+,
+G. Schnell
+, C. Schwanda
+, Y. Seino
+, K. Senyo
+, M. E. Sevior
+, W. Shan
+,
+M. Shapkin
+, C. Sharma
+, J.-G. Shiu
+, E. Solovieva
+, M. Stariˇc
+,
+M. Sumihama
+, T. Sumiyoshi
+, M. Takizawa
+, U. Tamponi
+, K. Tanida
+,
+F. Tenchini
+, M. Uchida
+, T. Uglov
+, Y. Unno
+, S. Uno
+, P. Urquijo
+,
+Y. Ushiroda
+, S. E. Vahsen
+, G. Varner
+, A. Vinokurova
+, D. Wang
+, E. Wang
+,
+M.-Z. Wang
+, X. L. Wang
+, S. Watanuki
+, X. Xu
+, B. D. Yabsley
+, W. Yan
+,
+S. B. Yang
+, J. Yelton
+, Y. Yook
+, L. Yuan
+, Y. Zhai
+, V. Zhilich
+,
+V. Zhukova
+,
+arXiv:2301.03768v1  [hep-ex]  10 Jan 2023
+
+Abstract: Charged-lepton-flavor-violation is predicted in several new physics scenarios.
+We update the analysis of τ lepton decays into a light charged lepton (ℓ = e± or µ±) and a
+vector meson (V 0 = ρ0, φ, ω, K∗0, or K∗0) using 980 fb−1 of data collected with the Belle
+detector at the KEKB collider. No significant excess of such signal events is observed, and
+thus 90% credibility level upper limits are set on the τ → ℓV 0 branching fractions in the
+range of (1.7–4.2) × 10−8. These limits are improved by 30% on average from the previous
+results.
+Keywords: e+–e− Experiments, Tau Physics
+
+Contents
+1
+Introduction
+1
+2
+Belle experiment
+1
+3
+Reconstruction and event selection
+2
+4
+Signal efficiency and background estimation
+6
+5
+Results
+10
+6
+Conclusion
+11
+1
+Introduction
+In the Standard Model, charged-lepton-flavor-violation (CLFV) is so strongly suppressed
+that it is undiscoverable by current experiments. Therefore, a discovery of a CLFV event
+indicates new physics (NP). Verifying various NP models requires many searches of various
+CLFV modes [1]. Whereas the CLFV constraints are much more stringent for µ-to-e than
+for τ through the precise measurements [2–4], we are interested in τ, the third-generation
+and heaviest lepton. So-called B-anomalies, which indicate NP effects in B semileptonic
+decays [5–16], also motivate the CLFV searches.
+We focus on τ CLFV decays into a charged lepton (ℓ = e± or µ±) and a neutral vector
+meson (V 0 = ρ0, φ, ω, K∗0, or K∗0). In refs. [17–22], the τ → µφ mode is a sensitive probe
+for leptoquark models that can explain the B-anomalies.1 Some other NP models predict
+branching fractions of O(10−10)–O(10−8) for τ → ℓV 0 [25–28].
+We previously searched for τ → ℓV 0 events using 854 fb−1 of Belle data, and set
+90% credibility level (C.L.) upper limits on the branching fractions in the range of (1.2–
+8.4) × 10−8 [29].2 This paper reports an updated search for τ → ℓV 0 using the full 980
+fb−1 Belle data set. The signal efficiency is improved through new event selection criteria
+with a multivariate analysis.
+2
+Belle experiment
+The Belle detector is a spectrometer that covers large solid angles of the e+e− collision
+events from the KEKB accelerator [30, 31]. The detector consists of a silicon vertex de-
+tector, a 50-layer central drift chamber (CDC), an array of aerogel threshold Cherenkov
+1One of the B-anomalies which motivated the models described in those references is the R(K(∗))
+anomaly reported by the LHCb experiment [23], but it disappeared in their updated analysis [24].
+2In common high energy physics usage, this credibility level has been reported as “confidence level,”
+which is a frequentist-statistics term.
+– 1 –
+
+counters, time-of-flight scintillation counters, and an electromagnetic calorimeter composed
+of 8736 CsI(Tl) crystals (ECL). These devices are located inside a superconducting solenoid
+coil that provides a 1.5 T magnetic field. An iron flux return located outside of the coil is
+instrumented to detect K0
+L mesons and identify muons. The Belle detector is described in
+detail elsewhere [32, 33].
+Of the 980 fb−1 data set, 703 fb−1 was collected at the Υ(4S) resonance, 121 fb−1 at
+the Υ(5S), 89 fb−1 at an energy 60 MeV below the Υ(4S), 28 fb−1 of energy-scans above the
+Υ(4S), and the remainder at and near the Υ(1–3S). Compared to the previous paper [29],
+the following data sets have been added: 78 fb−1 at and near the Υ(5S), 38 fb−1 at and
+near the Υ(1–3S), and 10 fb−1 at an energy 60 MeV below the Υ(4S).
+The e+e− collision events in the Belle detector are simulated by the Monte Carlo (MC)
+method. Signal MC events of τ → ℓV 0 are generated by a dedicated MC with KKMC and
+TAUOLA [34], where τ +τ − pairs are initially produced and one of the τ’s decays into ℓV 0
+and the other decays generically. The numbers of generated signal MC events are 1.1×106
+events at the Υ(4S) resonance, 0.4 × 106 events at the Υ(5S), 0.1 × 106 events at each of
+the Υ(1–3S), and 0.1 × 106 events at an energy 60 MeV below the Υ(4S). We assume a
+uniform CLFV decay angle in the τ rest frame. No specific NP model is assumed in the
+CLFV decay process, and the spin direction of V 0 is set randomly and independently of the
+spin of the mother τ. For background MC simulations, e+e− → q¯q (q = u, d, s, c), e+e− →
+τ +τ −, Bhabha, and two-photon processes are generated by EvtGen [35], KKMC [34],
+BHLUMI [36], and AAFH [37], respectively.
+The detector responses are simulated by
+GEANT3 [38].
+3
+Reconstruction and event selection
+A signal τ is reconstructed from a lepton and a neutral vector meson. We separate the
+event into two hemispheres in the center-of-mass (c.m.) frame by a plane perpendicular
+to the thrust vector (⃗nT ) [39, 40]. The thrust vector is obtained by maximizing the thrust
+T = Σi|⃗p c.m.
+i
+· ⃗nT |/Σi|⃗p c.m.
+i
+|, where i runs over all tracks and photons, and ⃗p c.m.
+i
+is the
+momentum in the c.m. frame. In the hemisphere that contains a τ CLFV decay (called
+signal side and τsig), V 0 is reconstructed as follows: ρ0 from π+π− within the reconstructed
+mass window of 0.445–1.08 GeV/c2, φ from K+K− within 1.00–1.04 GeV/c2, ω from
+π+π−π0 within 0.7–0.9 GeV/c2, K∗0 from K+π− within 0.7–1.1 GeV/c2, and K∗0 from
+K−π+ within 0.7–1.1 GeV/c2. In the other hemisphere (called tag side), the other τ (τtag)
+is reconstructed from ℓ±νν, π±ν, π±π0ν, π±π0π0ν, or π±π∓π±ν. This τtag information
+enables the suppression of background events that have no neutrinos in the tag side.
+The signal τ → ℓV 0 events have a unique kinematical feature; the ℓV 0 invariant mass
+(MℓV 0) is close to the τ mass and the difference of the ℓV 0 energy from the beam energy
+in the c.m. frame (∆E) is close to zero. The signal events within 1.65 GeV/c2 < MℓV 0 <
+1.90 GeV/c2 and |∆E| < 0.5 GeV are reconstructed in this paper.
+We follow a blind
+analysis approach in this search by not looking at the signal candidates in the data set
+until finalizing the event selection and background estimation. The blind region is 1.75
+– 2 –
+
+GeV/c2 ≤ MℓV 0 < 1.81 GeV/c2 and |∆E| < 0.08 GeV for the µρ0, µφ and µK∗0(K∗0)
+modes, and 1.74 GeV/c2 ≤ MℓV 0 < 1.82 GeV/c2 and |∆E| < 0.1 GeV for the other modes.
+Charged tracks, photons, and π0s should satisfy the following selection criteria. Each
+charged track or photon is within the fiducial volume defined by −0.866 < cos θ < 0.956,
+where θ is the polar angle with respect to the direction opposite to the e+ beam in the
+laboratory frame. Charged tracks come from the interaction point; the distance of the
+closest point from the interaction point is less than 0.5 cm in the transverse direction and
+less than 3.0 cm in the longitudinal direction. Each π0 is reconstructed from two photons
+inside the same hemisphere and the photon energy (Eγ) should be larger than 0.05 GeV.
+The π0 mass window is 0.12 GeV/c2 < Mγγ < 0.15 GeV/c2, corresponding to ±3σ in the π0
+mass resolution. A π0 mass-constrained fit is performed to improve the energy resolution.
+After reconstructing the signal and tag τ’s, no extra charged tracks are allowed. We
+count the number of photons (nγ) with Eγ larger than 0.1 GeV in the signal side, and
+require nγ ≤ 3 for the ℓω mode, which includes a π0 → γγ, and nγ ≤ 1 for the other
+modes.
+Particle identification is effective in suppressing the main background events of three-
+hadron-track to the τ → ℓV 0 signal. We use likelihood ratios for electron identification
+(P(e)) [41] and muon identification (P(µ)) [42].
+The lepton identification criteria are
+P(e) > 0.9 for electrons, and P(µ) > 0.95 and the momentum is larger than 0.6 GeV/c for
+muons. The electron (muon) identification efficiency is 90% (75%), whereas the probability
+of misidentifying a pion as an electron (muon) is 0.1% (2%). The energy loss of an electron
+by bremsstrahlung is recovered by adding back the energy of every photon within 0.05
+radians from the electron track direction into the electron momentum. To suppress low-
+multiplicity background events like Bhabha, ee → eeee, or ee → eeµµ, an electron veto
+(P(e) < 0.9) is applied to all hadron candidate tracks.
+For hadron identification, we use a binary likelihood ratio P(i|j) = Li/(Li+Lj), where
+Li(j) is the likelihood of particle i (j) [43] and i (j) is π, K, or p. The kaon identification
+criteria are P(K|π) > 0.6 for both charged kaons from φ decay and P(K|π) > 0.8 for the
+charged kaon from K∗0 and K∗0 decays. The kaon identification efficiency is 86% (77%),
+whereas the probability of misidentifying a charged pion as a kaon is 4% (2%) for the kaons
+from φ (K∗0, K∗0). A kaon veto (P(K|π) < 0.6) is applied to both charged pions from
+ρ0 in the signal side, and 96% of pions are retained, whereas 14% of kaons are not vetoed.
+To suppress muons from kaons decaying inside the CDC (K± → µ±ν), the kaon veto is
+also applied to the signal-side muon track for the hadronic tags (τtag → πν, ππ0ν, πππν,
+or ππ0π0ν). For the µV 0 modes with the hadronic tags, a proton veto (P(p|K) < 0.6 and
+P(p|π) < 0.6) is applied for the tag-side tracks.
+The signal events have one or two neutrinos from the τtag decay. We introduce some
+event selection criteria requiring one or more neutrinos in the tag side. The missing mo-
+mentum due to the neutrino(s) is calculated by subtracting the vector sum of the momenta
+of all tracks and photons from the sum of the beam momenta. The missing energy is also
+calculated by subtracting the sum of the energy of all tracks and photons from the sum
+of the beam energy. Here, extra photons that are not used for the τ reconstruction are
+included. The transverse missing momentum is required to be larger than 0.5 GeV/c, and
+– 3 –
+
+the missing energy in the c.m. frame (Ec.m.
+miss) is required to be larger than 0 GeV. Events
+with missing particles other than neutrinos should be rejected as background events. These
+non-neutrino missing particles can arise in two ways: neutral particles pass through the
+gaps between the barrel and end-cap ECLs, and any particles go outside the CDC volume.
+Thus, the direction of the missing momentum is required not to point to such regions. The
+missing particles should be in the tag side and hence cos θc.m.
+miss−tag > 0, where θc.m.
+miss−tag
+is the angle between the missing momentum and the vector sum of the momenta of the
+tag-side tracks and photons in the c.m. frame. The neutrino angle with respect to the τtag
+momentum direction is restricted in a τtag two-body decay; thus cos θc.m.
+miss−tag < 0.85 is also
+applied for the ℓρ0 modes with τtag → πν.
+We require features of a generic τ decay in the tag side. The invariant mass of the
+particles including all photons in the tag hemisphere should be less than the τ mass (1.777
+GeV/c2). For τtag decays into ππ0ν (3πν), the reconstructed mass of those pions is required
+to be 0.4 GeV/c2 < Mππ0 < 1.3 GeV/c2 (0.7 GeV/c2 < M3π < 1.7 GeV/c2), which
+corresponds to the mass of ρ± (a±
+1 ).
+After the above event reconstruction, the background sources are the q¯q continuum
+(q = u, d, s, c), generic τ +τ −, and low-multiplicity events.
+The low-multiplicity events
+especially contribute to the background events for eV 0 modes that have electron tracks.
+We suppress the low-multiplicity events first, and then use a maltivariate analysis tool to
+suppress the q¯q continuum and generic τ +τ − events.
+The Bhabha events have tracks from photon conversion. To suppress these background
+events for the eV 0 modes, the invariant mass of the electron and one of the tracks from the
+V 0, assigned the electron-mass hypothesis, should be larger than 0.2 GeV/c2. In addition,
+for the eK∗0 and eK∗0 modes, the invariant mass of the two tracks from the V 0, each
+assigned the electron-mass hypothesis, is required to be larger than 0.1 GeV/c2.
+This
+event selection also suppresses some of the generic τ +τ − events, which have tracks from
+photon conversion.
+The low-multiplicity background events are still not negligible for the events with elec-
+trons: τ → eV 0 or τtag → eνν. Because the missing particles of the low-multiplicity back-
+ground events are the bremsstrahlung photons from the electron in the tag side, cos θc.m.
+miss−tag
+is close to one (Figure 1). In addition, the missing energy is small for some low-multiplicity
+background events. For the µρ0 mode with τtag → eνν, cos θc.m.
+miss−tag < 0.99 and Ec.m.
+miss > 0.4
+GeV selection criteria are applied. For the eV 0 modes with τtag → eνν or πν, cos θc.m.
+miss−tag <
+0.97 is applied. For the eV 0 modes with τtag → eνν, Ec.m.
+miss should be larger than 0.4, 2.0,
+and 1.5 GeV for eφ, eρ0, and the other eV 0 modes, respectively.
+The remaining background events are mainly from the q¯q continuum (q = u, d, s, c)
+and generic τ +τ − events, which have three charged pion tracks in the signal side. We use a
+two-class Boosted Decision Tree (BDT) for signal and these background classification. The
+BDT library is LightGBM [44]. This BDT outputs a signal probability using the following
+input variables:
+• MV 0, M2
+ν , P c.m.
+ν
+, T, P sig
+ℓ , Ehemi
+tag , cos θc.m.
+miss−tag
+• (categorical variables) τtag decay mode, collision energy
+– 4 –
+
+Figure 1: The cos θc.m.
+miss−tag distribution of the τ → eρ0 mode with a electron tag track
+after the reconstruction, particle identification, and photon conversion event suppression.
+Black points with error bars are the data outside the blind region. Red solid histogram
+is the signal MC. The signal MC is scaled to the number of events corresponding to 100
+times as large branching fraction as the current upper limit. The red dashed line is the
+upper limit to remove the low-multiplicity events.
+The low-multiplicity events cluster
+around cos θc.m.
+miss−tag = 1, whereas the other background events are linearly distributed in
+the region of cos θc.m.
+miss−tag > 0.8.
+• (additional for the ℓω modes) P sig
+π0 , Elow
+γ
+,
+where MV 0 is the invariant mass of the vector meson, M2
+ν is the missing mass squared, P c.m.
+ν
+is the missing momentum in the c.m. frame, T is the magnitude of the thrust vector [39, 40],
+P sig
+ℓ
+is the momentum of the lepton in the signal side, Ehemi
+tag
+is the energy sum of the tracks
+and photons in the tag hemisphere, P sig
+π0 is the momentum of π0 from ω and Elow
+γ
+is the
+lower energy of the two photons from the π0. The variables of neutrino kinematics (M2
+ν and
+P c.m.
+ν
+) were not used for the event selection in the previous paper [29]. They are calculated
+from the momenta of the reconstructed τsig and τtag, where the energy of τsig is fixed to the
+half of the beam energy in the c.m. frame. The q¯q continuum background events can be
+effectively suppressed by a M2
+ν selection in the hadronic tags, involving only one neutrino
+(Figure 2).
+The training, validation and evaluation of the BDT are done with 40%, 10%, and
+50% of the signal MC, respectively. Regarding the training and validation samples for
+the background events, we utilize hadron background enhanced data that are obtained by
+removing the lepton identification for the signal-side leptons but with a lepton identification
+veto (P(e) ≤ 0.9 and P(µ) ≤ 0.95) for all the signal-side tracks in the data. The hadron
+background enhanced data have a much larger number of events than the background data
+with the nominal selection criteria, whereas both data sets are composed mainly of three
+charged pions from τ decays or from continuum events. The training is done with 80%
+of the hadron background enhanced data and the validation is done with 20%. During
+BDT training, a weight is applied to each of the signal MC events such that the sum of
+the weights is equal to the number of background events.
+We monitor the area under
+– 5 –
+
+t→μpo, electron tag
+70
+BR(→μp0)=1.2×10-8 × 100
+Number of events/(0.01)
+60
+data
+50
+40
+30
+20
+10
+t+.+-
+0.2
+0.0
+0.4
+0.6
+0.8
+1.0
+CosAc.m.
+miss -tagcurve (AUC) of the Receiver Operating Characteristic curve [45] for the validation samples
+during the training and choose the BDT with the best AUC score.
+The event selection with the BDT output (BDT selection) is determined only by a
+target signal efficiency.
+The target signal efficiency is determined based on the signal
+efficiency with a cut-based event selection.
+In the cut-based event selection, the MV 0
+windows correspond to ±2σ of reconstructed mass distribution, and the M2
+ν windows are
+set for each ℓV 0 mode and each τtag decay mode so that the expected number of background
+events inside the signal region (NBG, see the next section) is approximately one or less.
+The target signal efficiency with the BDT selection is set as relatively 5% larger than that
+with the cut-based event selection, because we expect improvement in separating the signal
+events from the background events.
+The finalized BDT selection shows similar NBG to that of the cut-based event selection.
+The BDT selection is not applied to the ℓφ modes because NBG in each of the two modes
+is small enough.
+Figure 2: The M2
+ν distribution of the τ → µρ0 mode with the hadronic tags after the
+event selection except for the requirement of the BDT output. Black points with error bars
+are the data outside the blind region. Red solid histogram is the signal MC. The signal MC
+is scaled to the number of events corresponding to 100 times as large a branching fraction
+as the current upper limit. The events constituting the upper tail of the signal distribution
+originate from wrong or missing π0 in the tag side.
+4
+Signal efficiency and background estimation
+We define the signal region with an ellipse in the MℓV 0–∆E plane. Most of the signal
+events cluster around MℓV 0 = 1.777 GeV/c2 and ∆E = 0 GeV with some correlation. The
+ellipse oblateness and the rotation angle are calculated from the covariance matrix of the
+signal MC distribition after the event selection. The center of the ellipse is the mean of the
+distribution. The ellipse size is determined to maximize the figure-of-merit (FOM) [46],
+FOM =
+ε
+α
+2 + √NBG
+,
+(4.1)
+– 6 –
+
+t→μpo, hadronic tag
+300
+BR(t→μp°)=l.2×10-8 × 100
+250
+data
+200
+150
+100
+50
+-2
+-1
+0
+1
+2
+M2 (GeV2/c4)where ε is the signal efficiency inside the ellipse, α is the confidence coefficient (α = 1.64
+at 90% C.L.).
+Figure 3: The MℓV 0 vs. ∆E distribution of the τ → µρ0 hadron background enhanced
+samples: the data (upper side), the generic τ +τ − MC (lower left) and the q¯q continuum
+MC (lower right, q = u, d, s, c). The range of the ∆E axis is limited to the fitting region.
+The MC sets are scaled to the data. The low-multiplicity background events are negligible
+for the hadron background enhanced samples and are not shown in this figure.
+We estimate NBG through interpolation from the sideband data. Here the sideband
+data is a set of data passing the event selection and inside the sideband region: 1.65 GeV/c2
+< MℓV 0 < 1.9 GeV/c2 and |∆E| < 0.1 GeV outside of the blind region. The interpolation
+is based on a function in the MℓV 0–∆E plane. This function is obtained by fitting the
+distribution of the hadron background enhanced data within |∆E| < 0.1 GeV, and then it
+is scaled to the sideband data. Figure 3 shows the distributions of the hadron background
+enhanced data and MC for the µρ0 mode. The function is:
+F(MℓV 0, ∆E) = f(MℓV 0) ×
+1
+1 + exp[ay(∆E − y0)] + cflat
+0 ,
+(4.2)
+– 7 –
+
+data
+0.100
+100
+0.075
+0.050
+80
+△E (GeV)
+0.025
+60
+0.000
+-0.025
+40
+-0.050
+20
+-0.075
+-0.100
+0
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mμpo (GeV/c2)t+t- MC
+0.100
+0.075
+80
+0.050
+△E (GeV)
+60
+0.025
+0.000
+40
+-0.025
+-0.050
+20
+-0.075
+-0.100
+0
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mμpo (GeV/c2)qq MC (q = u, d,s,c)
+0.100
+8
+0.075
+7
+0.050
+6
+(GeV)
+0.025
+5
+0.000
+4
+△E 
+-0.025
+3
+-0.050
+2
+-0.075
+1
+-0.100
+0
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mμpo (GeV/c2)f(x) =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+� x+5σ
+x−5σ
+g(x′) ×
+1
+√
+2πσexp
+�−(x − x′)2
+2σ2
+�
+dx′
+(V 0 = ρ0, ω)
+c1(x − x0)2 + c0
+(V 0 = K∗0, K∗0)
+c0
+(V 0 = φ)
+g(x) =
+�
+�
+�
+�
+�
+c1[(x − x0)2 + k(x − x0)] + c0
+(x < x0, V 0 = ρ0)
+c1(x − x0) + c0
+(x < x0, V 0 = ω)
+c0
+(x ≥ x0)
+(4.3)
+where f(x) represents the background distribution as a function of MℓV 0; c1, c0, x0, and k
+are parameters that define the shape of the function; ay represents sharpness of the sigmoid
+function along the ∆E axis; y0 is the center of the sigmoid function; and cflat
+0
+is a term of
+flat background events in the MℓV 0–∆E plane. We define f(x) for each V 0 in eq. (4.3) and
+the functions for the ℓρ0 (ℓω) modes are smeared by a Gaussian with standard deviation
+(σ) of 6.6 (9.6) MeV/c2. This σ corresponds to the mass resolution that affects the edge
+of the MℓV 0 distribution close to the τ mass for the τ +τ − background. The edge is broad
+for the other modes owing to wrong mass assignment of fake kaons. The τ +τ − background
+events for the ℓφ modes are included in c0 because they are flat along the MℓV 0 axis in
+1.65 GeV/c2 < MℓV 0 < 1.9 GeV/c2.
+We obtain the optimal fit parameters by a likelihood fit using MINUIT [47].
+The
+following region is excluded from the fitting to avoid D+ → K−π+π+ and D+ → π+φ
+background events, which cluster around the D meson mass: 1.83(1.82) GeV/c2 ≤ MℓV 0 <
+1.89 GeV/c2 and ∆E < 0.04 GeV for the µK∗0 (eK∗0) and µφ (eφ) modes.
+The parameters of ay, y0, k, and x0 are fixed at the fit results of the hadron background
+enhanced data within |∆E| < 0.1 GeV. The fit uncertainties of these fixed parameters are
+included in the systematic uncertainty of NBG. The other fit parameters correspond to
+the scale factors of each background component: generic τ +τ − (c1), and continuum and
+low-multiplicity background events (c0 and cflat
+0 ). We fit the function floating these scale
+factors (c1, c0, and cflat
+0 ) to the sideband data. The same region around the D meson mass
+as for the fit to the hadron background enhanced data is excluded from the fitting for the
+ℓφ and ℓK∗0 modes. The functions are integrated in the elliptical signal regions to deduce
+NBG, which are in the range of 0.25–0.95.
+Another systematic uncertainty on NBG comes from difference of the MℓV 0–∆E dis-
+tributions between the sideband data and the hadron background enhanced data within
+|∆E| < 0.1 GeV. The difference mainly arises from the electron (muon) identification
+fake rate, Rfake
+e(µ)(P, θ), which depends on the momentum P and θ of the track. The side-
+band data have a pion misidentified as a lepton, which tends to have a lower momen-
+tum than the pions in the hadron background enhanced data. We evaluate a change of
+NBG when the parameters—ay, y0, k, and x0—are redetermined with weighted hadron
+background enhanced data, where each event is weighted by the ratio of Rfake
+e(µ)(P, θ) to
+1 − Rfake
+e
+(P, θ) − Rfake
+µ
+(P, θ) for the track in order to conform the MℓV 0–∆E distribution
+to the one of the sideband data. The amout of change of NBG is taken as the systematic
+uncertainty of NBG.
+The statistical uncertainty of NBG is calculated as follows: We generate 100 sets of
+– 8 –
+
+pseudo-data for each mode in the MℓV 0–∆E histogram. The content of each bin in the
+histogram is set randomly following a Poisson distribution, with the mean taken from the
+function fitted to the sideband data. We fit the function to each set of the pseudo-data to
+deduce NBG, and the standard deviation of these NBG is taken as the statistical uncertainty.
+The major contribution to NBG comes from the MℓV 0 flat term in eq. (4.2) (c0 and
+cflat
+0 ), which corresponds to the continuum or low-multiplicity background events.
+The
+contribution of the generic τ +τ − background events, which depends on MℓV 0, is about
+one-third as large as the other background contributions. We cannot distinguish the back-
+ground components of the ℓφ modes through the fit to the data, because the generic τ +τ −
+background events are distributed evenly along the MℓV 0 axis.
+The systematic uncertainties of the expected number of signal events are listed in
+Table 1. The dominant uncertainties are from the particle identification.
+The track and photon energy resolutions in the MC are corrected such that the
+mass resolution of the D(∗)+ meson matches between the data and MC, where D(∗)+ →
+K−π+π+(π0) is reconstructed with similar event selection criteria to the signal ones (e.g.
+|∆E| < 0.5 GeV). The uncertainty of the data mass resolution propagates to the uncer-
+tainties of the corrected energy resolutions. We generate two additional signal MC sets in
+which the track (photon) energy resolution is different by plus and minus its uncertainty,
+and take the half of the difference in the expected number of the signal events as the
+systematic uncertainty.
+All the uncertainties in Table 1 are summed in quadrature to yield the total systematic
+uncertainties shown in Table 2.
+Table 1: List of the systematic uncertainties of the expected number of signal events. The
+average number of tracks (particles) in the reconstructed τ +τ − events for each signal mode
+is represented as Ntrack(particle). When the uncertainty is different mode by mode, we show
+the range of the uncertainty.
+Source
+σsyst (%)
+Integrated luminosity
+1.4
+ee → ττ(γ) cross section [48]
+0.3
+B(φ → KK) and B(ω → πππ0)
+1.2 and 0.7
+Trigger efficiency
+0.2–0.9
+Tracking efficiency
+0.35 × Ntrack
+Electron identification efficiency
+1.7 × Nelectron
+Muon identification efficiency
+1.8 × Nmuon
+K and π identification efficiency
+1.6 (ρ0), 1.8 (φ) and 1.1 (K∗0 and K∗0)
+π0 efficiency
+2.2 × Nπ0
+Electron veto for hadrons
+0.4–1.2
+MC statistics
+0.3–0.5
+Track energy resolution
+0.3–1.3
+Photon energy resolution
+0.0–0.4
+– 9 –
+
+5
+Results
+Figures 4 and 5 show the observed event distributions in the MℓV 0–∆E plane. The observed
+number of events in the signal region (Nobs) has no excess over NBG.
+We set 90% C.L. upper limits on the branching fractions based on a Bayesian method
+with the use of Markov Chain Monte Carlo [49].
+The probability density function of
+the branching fraction (B(τ → ℓV 0)) is calculated assuming that Nobs follows a Poisson
+distribution function whose mean value is the expected number of events (Nexp),
+Nexp = L × 2σττB(τ → ℓV 0) × ε + NBG,
+(5.1)
+where L is the integrated luminosity (980.4 ± 13.7 fb−1), σττ is the cross section of τ-pair
+production that is calculated with KKMC [48] (the weighted average of σττ at all the beam
+energies is 0.916 ± 0.003 nb), and ε is the signal efficiency including the branching fraction
+of the V 0. We assume that these values (L, σττ, ε, and NBG) follow a Gaussian distribution
+with the width equal to the uncertainty of each value.
+The upper limits on B(τ → ℓV 0) are listed in Table 2. The average of the limits is
+better than that of the previous results using 854 fb−1 [29] by 30%. This is due to the
+additional 15% of integrated luminosity; the addition of π±π∓π±ν and π±π0π0ν modes in
+τtag reconstruction, which increases the signal efficiency by 9.6%; and the event selection
+by multivariate analysis (BDT). The upper limit on B(τ → µρ0) is worse than that of
+the previous result, though the expected upper limit before unblinding is better. This is
+because we use the Bayesian limits instead of the Frequentist limits, which are negatively
+proportional to NBG when Nobs is fixed.
+Table 2: The signal efficiency (ε), the expected number of background events (NBG),
+total systematic uncertainty of the expected number of signal events (σsyst), the number
+of observed events in the signal region (Nobs), and the observed 90% C.L. upper limits on
+the branching fraction (Bobs (10−8)).
+Mode
+ε (%)
+NBG
+σsyst (%)
+Nobs
+Bobs (×10−8)
+τ − → µ−ρ0
+7.78
+0.95±0.20(stat.) ±0.11(syst.)
+4.6
+0
+< 1.7
+τ − → e−ρ0
+8.49
+0.80±0.27(stat.) ±0.02(syst.)
+4.4
+1
+< 2.2
+τ − → µ−φ
+5.59
+0.47±0.15(stat.) ±0.05(syst.)
+4.8
+0
+< 2.3
+τ − → e−φ
+6.45
+0.38±0.21(stat.) ±0.00(syst.)
+4.5
+0
+< 2.0
+τ − → µ−ω
+3.27
+0.32±0.23(stat.) ±0.03(syst.)
+4.8
+0
+< 3.9
+τ − → e−ω
+5.41
+0.74±0.43(stat.) ±0.01(syst.)
+4.5
+0
+< 2.4
+τ − → µ−K∗0
+4.52
+0.84±0.25(stat.) ±0.03(syst.)
+4.3
+0
+< 2.9
+τ − → e−K∗0
+6.94
+0.54±0.21(stat.) ±0.12(syst.)
+4.1
+0
+< 1.9
+τ − → µ−K∗0
+4.58
+0.58±0.17(stat.) ±0.06(syst.)
+4.3
+1
+< 4.2
+τ − → e−K∗0
+7.45
+0.25±0.11(stat.) ±0.01(syst.)
+4.1
+0
+< 1.7
+– 10 –
+
+6
+Conclusion
+To conclude, we searched for lepton-flavor-violating τ decays into one lepton and one vector
+meson using the full 980 fb−1 of Belle data. No statistically significant signal candidates are
+observed, and the 90% C.L. upper limits on the branching fraction are in the range of (1.7–
+4.2) × 10−8 for τ → µV 0 and (1.7–2.4) × 10−8 for τ → eV 0. The upper limits are improved
+by 30% on average from the previous results. We achieve these improvements both with
+the reconsideration of the event selection criteria and with the 126 fb−1 of additional data
+set.
+Acknowledgments
+This work, based on data collected using the Belle detector, which was operated until
+June 2010, was supported by the Ministry of Education, Culture, Sports, Science, and
+Technology (MEXT) of Japan, the Japan Society for the Promotion of Science (JSPS),
+and the Tau-Lepton Physics Research Center of Nagoya University; the Australian Re-
+search Council including grants DP180102629, DP170102389, DP170102204, DE220100462,
+DP150103061, FT130100303; Austrian Federal Ministry of Education, Science and Re-
+search (FWF) and FWF Austrian Science Fund No. P 31361-N36; the National Natural
+Science Foundation of China under Contracts No. 11675166, No. 11705209; No. 11975076;
+No. 12135005; No. 12175041; No. 12161141008; Key Research Program of Frontier Sci-
+ences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; Project
+ZR2022JQ02 supported by Shandong Provincial Natural Science Foundation; the Ministry
+of Education, Youth and Sports of the Czech Republic under Contract No. LTT17020;
+the Czech Science Foundation Grant No. 22-18469S; Horizon 2020 ERC Advanced Grant
+No. 884719 and ERC Starting Grant No. 947006 “InterLeptons” (European Union); the
+Carl Zeiss Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster Uni-
+verse, and the VolkswagenStiftung; the Department of Atomic Energy (Project Identi-
+fication No.
+RTI 4002) and the Department of Science and Technology of India; the
+Istituto Nazionale di Fisica Nucleare of Italy; National Research Foundation (NRF) of
+Korea Grant Nos. 2016R1D1A1B02012900, 2018R1A2B3003643, 2018R1A6A1A06024970,
+RS202200197659, 2019R1I1A3A01058933, 2021R1A6A1A03043957, 2021R1F1A1060423,
+2021R1F1A1064008, 2021R1A4A2001897, 2022R1A2C1003993; Radiation Science Research
+Institute, Foreign Large-size Research Facility Application Supporting project, the Global
+Science Experimental Data Hub Center of the Korea Institute of Science and Technology
+Information and KREONET/GLORIAD; the Polish Ministry of Science and Higher Ed-
+ucation and the National Science Center; the Ministry of Science and Higher Education
+of the Russian Federation, Agreement 14.W03.31.0026, and the HSE University Basic Re-
+search Program, Moscow; University of Tabuk research grants S-1440-0321, S-0256-1438,
+and S-0280-1439 (Saudi Arabia); the Slovenian Research Agency Grant Nos. J1-9124 and
+P1-0135; Ikerbasque, Basque Foundation for Science, Spain; the Swiss National Science
+Foundation; the Ministry of Education and the Ministry of Science and Technology of Tai-
+wan; and the United States Department of Energy and the National Science Foundation.
+– 11 –
+
+These acknowledgements are not to be interpreted as an endorsement of any statement
+made by any of our institutes, funding agencies, governments, or their representatives. We
+thank the KEKB group for the excellent operation of the accelerator; the KEK cryogenics
+group for the efficient operation of the solenoid; and the KEK computer group and the
+Pacific Northwest National Laboratory (PNNL) Environmental Molecular Sciences Labora-
+tory (EMSL) computing group for strong computing support; and the National Institute of
+Informatics, and Science Information NETwork 6 (SINET6) for valuable network support.
+– 12 –
+
+(a) τ → µρ0
+(b) τ → µφ
+(c) τ → µω
+(d) τ → µK∗0
+(e) τ → µK∗0
+Figure 4: Observed event distributions of MℓV 0 vs. ∆E after the τ → µV 0 event selection.
+Black points are the data, blue squares show the signal MC distribution with an arbitrary
+normalization. The red ellipse line is the signal region. The estimation of the number of
+background events is done using the data between the red horizontal lines except the blind
+region.
+– 13 –
+
+0.4
+口
+Data
+0.2
+△E (GeV)
+0.0
+-0.2
+-0.4
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Muk*0 (GeV/c2)0.4
+口
+Data
+0.2
+(GeV)
+0.0
+△E (
+-0.2
+-0.4
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mμpo (GeV/c2)口
+0.4
+Data
+0.2
+(GeV)
+0.0
+△E (
+-0.2
+:
+-0.4
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mus (GeV/c2)3n↑
+口
+0.4
+Data
+0.2
+△E (GeV)
+0.0
+-0.2
+-0.4
+1.70
+1.65
+1.75
+1.80
+1.85
+1.90
+Mμw (GeV/c2)0.4
+口
+Data
+0.2
+(GeV)
+0.0
+△E (
+.
+-0.2
+。
+..
+-0.4
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Muk*0 (GeV/c2)(a) τ → eρ0
+(b) τ → eφ
+(c) τ → eω
+(d) τ → eK∗0
+(e) τ → eK∗0
+Figure 5: Observed event distributions of MℓV 0 vs. ∆E after the τ → eV 0 event selection.
+Black points are the data, blue squares show the signal MC distribution with an arbitrary
+normalization. The red ellipse line is the signal region. The estimation of the number of
+background events is done using the data between the red horizontal lines except the blind
+region.
+– 14 –
+
+0.4
+口
+Data
+0.2
+(GeV)
+0.0
+△E (
+-0.2
+-0.4
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mepo (GeVc2)t→ed
+口
+0.4
+Data
+0.2
+(GeV)
+0.0
+△E (
+-0.2
+-0.4
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mes (GeV/c²)ma↑
+0.4
+Data
+0.2
+E (GeV)
+0.0
+V
+-0.2
+-0.4
+1.65
+1.70
+1.75
+1.80
+1.85
+1.90
+Mew (GeV/c2)T→ek*0
+口
+0.4
+Data
+0.2
+(GeV)
+0.0
+E(
+-0.2
+-0.4
+1.80
+1.85
+1.65
+1.70
+1.75
+1.90
+Mek* (GeV/c2)T→ek*0
+口
+0.4
+Data
+0.2
+E (GeV)
+0.0
+V
+-0.2
+-0.4
+1.65
+1.75
+1.80
+1.85
+1.90
+1.70
+Mek* (GeV/c2)References
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+Principal deuterium Hugoniot via Quantum Monte Carlo and ∆-learning
+Giacomo Tenti,1, ∗ Andrea Tirelli,1, † Kousuke Nakano,1, 2, ‡ Michele Casula,3 and Sandro Sorella1, 4
+1International School for Advanced Studies (SISSA), Via Bonomea 265, 34136 Trieste, Italy
+2School of Information Science, JAIST, Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan
+3Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie (IMPMC),
+Sorbonne Universit´e, CNRS UMR 7590, MNHN, 4 Place Jussieu, 75252 Paris, France
+4Computational Materials Science Research Team,
+RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo 650-0047, Japan
+(Dated: January 10, 2023)
+We present a study of the principal deuterium Hugoniot for pressures up to 150 GPa, using
+Machine Learning potentials (MLPs) trained with Quantum Monte Carlo (QMC) energies, forces
+and pressures. In particular, we adopted a recently proposed workflow based on the combination
+of Gaussian kernel regression and ∆-learning.
+By fully taking advantage of this method, we
+explicitly considered finite-temperature electrons in the dynamics, whose effects are highly relevant
+for temperatures above 10 kK. The Hugoniot curve obtained by our MLPs shows an excellent
+agreement with the most recent experiments, with an accuracy comparable to the best DFT
+functionals. Our work demonstrates that QMC can be successfully combined with ∆-learning to
+deploy reliable MLPs for complex extended systems across different thermodynamic conditions, by
+keeping the QMC precision at the computational cost of a mean-field calculation.
+Introduction −
+The study of hydrogen under extreme
+conditions has been a very active topic in condensed
+matter physics. Hydrogen is the most abundant element
+in the universe and the accurate knowledge of its phase
+diagram at pressures of the order of hundreds of GPa is
+extremely important for a variety of applications, such
+as modelling the interior of stars and giant gas planets
+[1–3], the inertial-confinement fusion [4], and the high-
+Tc hydrogen-based superconductors [5, 6]. Nevertheless,
+several properties of this system are still highly debated,
+even at the qualitative level [7–10].
+One of the main reasons that hamper our full
+understanding of high-pressure hydrogen is the difficulty
+of reproducing extreme pressures in a laboratory. Typical
+shock-wave experiments [11] make use of accelerated flyer
+plates to compress a material sample in a very short time,
+thus allowing to study the specimen at high temperatures
+and pressures.
+In particular, the set of possible end-
+states that the system can reach from some given initial
+conditions, also named principal Hugoniot, must satisfy
+a set of equations, known as Rankine-Hugoniot (RH)
+relations [12], linking the thermodynamic properties of
+the final shocked state with those of the starting one.
+During the years, the principal deuterium Hugoniot has
+been measured for a wide range of pressures and with a
+great degree of accuracy [13–20], reaching a relative error
+on the density as small as 2% in recent experiments.
+In this context, numerical approaches, - in particular
+Ab Initio Molecular Dynamics (AIMD) simulations -,
+are extremely valuable, since they are not constrained
+by any experimental setup and can thus give further
+insight into this part of the phase diagram [21].
+The
+Hugoniot region is particularly important because of
+the availability of experimental data that can be used
+to benchmark different theoretical methods.
+Among
+them, Density Functional Theory (DFT) simulations
+have been extensively used and provided excellent
+results for the Hugoniot curve [22–28].
+However,
+the approximations behind the particular exchange-
+correlation functional often produce discrepancies across
+existing DFT schemes whose accuracy varies according to
+the thermodynamic conditions, making the functional-
+based approach unsatisfactory. Quantum Monte Carlo
+(QMC) simulations, which depend on more controllable
+approximations,
+have also been performed [29, 30].
+Although in principle more accurate and systematically
+improvable,
+these calculations have a much larger
+computational cost than DFT, and they are thus
+limited in system size and simulation length. Moreover,
+previous QMC calculations seem to give results for
+the principal Hugoniot in disagreement with the most
+recent experimental data, with the possible origin of this
+discrepancy being recently debated [31].
+To overcome the large computational cost of ab
+initio simulations, machine learning techniques, aimed
+at constructing accurate potential energy surfaces, have
+become increasingly popular. Within this approach, one
+uses a dataset of configurations, i.e.
+the training set,
+to build a machine learning potential (MLP) that is
+able to reproduce energies and forces calculated with
+the given target method [32].
+Unlike DFT MLPs,
+the QMC ones are relatively less common, given the
+larger computational cost and the consequent difficulty of
+generating large datasets, usually necessary to construct
+accurate MLPs.
+In this work, we have successfully built a very accurate
+MLP with QMC energies, forces and pressures in the
+region of the deuterium Hugoniot, using the so-called ∆-
+learning approach. The Hugoniot curve computed by the
+MLP shows an excellent agreement with the most recent
+arXiv:2301.03570v1  [cond-mat.str-el]  9 Jan 2023
+
+2
+experiments, and it shares with the best DFT functionals
+the same, - if not better -, accuracy.
+Computational details −
+In order to build an MLP
+with QMC references, we employed a combination of
+Gaussian Kernel Regression (GKR), Smooth Overlap
+of Atomic Positions (SOAP) descriptors [33], and ∆-
+learning. The same approach has been recently proposed
+in Ref. 34, where it was applied to the study of high-
+pressure hydrogen in similar thermodynamic conditions.
+Following the ∆-learning approach, an MLP is trained
+on the difference between the target method and a
+usually much cheaper baseline potential.
+Here, we
+trained 5 different MLPs,
+using Variational Monte
+Carlo (VMC) and Lattice Regularized Diffusion Monte
+Carlo (LRDMC) [35, 36] datapoints as targets, and
+several DFT baselines, with the Perdew-Zunger Local
+Density Approximation (PZ-LDA) [37], the Perdew-
+Burke-Ernzerhof (PBE) [38] and the van der Waals
+(vdW) -DF [39, 40] functionals. The QMC calculations
+were performed using the TurboRVB package [41].
+To determine the principal Hugoniot, we made use of
+the RH jump equation:
+H(ρ, T) = e(ρ, T) − e0 + 1
+2(ρ−1 − ρ−1
+0 ) [p(ρ, T) + p0] = 0,
+(1)
+where ρ,
+T,
+e(ρ, T),
+p(ρ, T) and ρ0,
+T0,
+e0,
+p0
+are the density, temperature, energy per particle and
+pressure of the final and initial states, respectively. In
+particular, we ran a first set of NV T simulations at
+several temperatures in the [4 kK : 10 kK] range, and
+Wigner-Seitz radii between 1.80 Bohr and 2.28 Bohr,
+corresponding to the range where the zero of H(ρ, T) was
+expected. These simulations were performed considering
+classical nuclei and ground-state electrons, as quantum
+corrections and thermal effects have been shown to be
+negligible for these temperatures [30]. At each step, the
+energy, forces and pressure were calculated using the
+Quantum Espresso package in its GPU accelerated
+version [42–44] with the chosen functional (PBE in most
+cases), and then corrected with our MLP trained on the
+difference between QMC and DFT data. The resulting
+dynamics has the same efficiency as a standard DFT
+AIMD simulation, which is roughly 100 times faster
+than the original QMC one.
+The details of our QMC
+simulations are reported in the Supplemental Material
+(SM) [45].
+For the DFT simulations, we considered a
+60 Ry plane-wave cutoff with a Projector Augmented
+Wave (PAW) pseudopotential [46] and a 4 × 4 × 4
+Monkhorst-Pack k-point grid, while for the dynamics we
+used a time step of 0.25 fs and a Langevin thermostat [47,
+48] with damping γ = 13 ps−1. For each temperature, the
+Hugoniot (ρ∗, p∗) coordinates are determined by fitting
+the Hugoniot function H(ρ, T) and the pressure p(ρ, T)
+with a spline function, and by numerically finding ρ∗ and
+the corresponding p∗.
+Within our approach, we can fully take advantage
+of the ∆-learning method by estimating the effect
+of thermalized electrons in our calculations.
+To do
+so, we considered two MLPs trained on the VMC-
+LDA and LRDMC-LDA differences, respectively, and
+ran simulations at temperatures T = 10 kK, 15 kK,
+and 35 kK with the corrected Karasiev-Sjostrom-Dufty-
+Trickey (KSDT) finite-temperature (FT) LDA functional
+[49–51] as baseline, in place of the usual ground-state PZ-
+LDA functional. In this way, we can include the effects of
+thermally excited electrons in our MLP without changing
+it, at least at the DFT level of theory.
+Results and Discussion −
+Fig. 1a shows our results
+together with several experimental values for pressures
+below 150 GPa [16, 19, 20]. We also report the principal
+Hugoniot obtained by directly using the PBE baseline,
+and the Coupled Electron Ion Monte Carlo (CEIMC)
+results of Ref. 30 for comparison. For T = 10 kK we
+show both the ground-state and FT results obtained
+with the procedure described previously, while for larger
+temperatures we plotted only the latter. Both the VMC
+and LRDMC models seem to reproduce very accurately
+the experimental points over the entire range of pressure
+considered.
+With respect to the most accurate data
+of Ref. 19, our estimate of the relative density ρ/ρ0 at
+the compressibility peak is only 1% lower for the VMC
+model and 3% lower for the LRDMC model, both being
+compatible within one error bar.
+Our results are in
+better agreement with experiments than the CEIMC ones
+reported in Ref. 30, which predicts a relative density 10%
+larger for the Hugoniot curve. The disagreement between
+the two results seems to be due to a large difference in the
+pressure estimates between the two methods, as further
+discussed in the SM [45].
+Fig. 1b displays the same points in the up − Us space,
+where up is the particle velocity and Us is the shock
+velocity, the two being calculated using the following RH
+relations:
+up =
+�
+(p + p0)(ρ−1
+0
+− ρ−1),
+Us = ρ−1
+0
+�
+p + p0
+ρ−1
+0
+− ρ−1 .
+The difference ∆Us between these points and the linear
+fit on the gas-gun data re-analyzed in Ref. 19 is also
+shown (bottom panel of Fig. 1b). Notice that the drop
+in the slope of Us relative to up coincides with the
+onset of the molecular-atomic (MA) transition, while the
+magnitude of the ∆Us minimum relates to the position
+of the relative compression peak.
+In particular, the
+PBE Hugoniot curve manifests a premature start of the
+dissociation, while it predicts correctly the magnitude of
+the compressibility maximum.
+Remarkably, our QMC
+results are very similar to the experimental findings not
+
+3
+2.5
+3.0
+3.5
+4.0
+4.5
+5.0
+/
+0
+0
+20
+40
+60
+80
+100
+120
+140
+160
+Pressure (GPa)
+PBE-FT
+VdW-DF1 (Ref.19)
+Experiments
+VMC
+VMC-FT
+LRDMC
+LRDMC-FT
+VMC (Ref. 30)
+RMC (Ref. 30)
+(a)
+15
+20
+25
+30
+35
+Shock velocity Us (km/s) 
+10
+15
+20
+25
+Particle velocity up (km/s)
+0.0
+0.5
+1.0
+1.5
+Us (km/s)
+(b)
+FIG. 1: (1a) Principal Hugoniot in the density-pressure space. Red and yellow circles are the results obtained with our
+MLPs trained on VMC and LRDMC datapoints, respectively, and a PBE baseline. Empty symbols refer to the results
+obtained using the finite-temperature (FT) KSDT functional as baseline. Blue and pink triangles are the PBE result
+calculated in this work and the VdW-DF1 result of Ref. 19 respectively. CEIMC results of Ref. 30 based on Variational
+Monte Carlo (VMC) and Reptation Monte Carlo (RMC) are reported in green squares. Cyan diamonds are the experimental
+results of Refs. 16, 19, and 20. Dashed-dotted lines are guides for the eye. (1b) [top panel] Hugoniot in the up–Us space.
+Black-dashed line is the re-analyzed gas-gun fit reported in Ref. 19. [bottom panel] Relative shock velocity with respect to the
+gas-gun fit. Only the experimental points of Ref. 19 are reported.
+only for the compressibility peak but also for the shock
+velocity slope.
+Thus, the Hugoniot curve obtained by our MLPs
+shows a much better agreement with the most recent
+experiments than the PBE functional, and is close to
+improved functionals, such as VdW-DF1 reported in Fig.
+1, which has been proved more accurate than PBE for
+high pressure hydrogen [52]. Cancellation of errors taking
+place in the DFT Hugoniot [31] is less apparent in the
+∆Us = ∆Us(up) relation (Fig. 1b), where the difference
+between PBE and improved theories is clear.
+The presence of an MA transition is also investigated
+in Fig. 2,
+where we report the radial distribution
+function, g(r), calculated on trajectories obtained with
+the LRDMC model for several temperatures at densities
+close to the Hugoniot curve. The inset of Fig. 2 displays
+the value of the molecular fraction m, defined as the
+percentage of atoms that stay within a distance of 2 Bohr
+(roughly corresponding to the first g(r) minimum after
+the molecular peak) from another particle for longerthan
+a characteristic time, here set to 6 fs. The results show
+a distinct atomic character for T ≥ 10 kK and a clear
+molecular peak at lower temperatures.
+Error analysis −
+To
+assess
+the
+quality
+of
+our
+principal Hugoniot determination,
+we analyzed the
+possible sources of errors in relation to our machine
+learning scheme. There are three main sources of errors:
+the uncertainties in the fit of H(ρ, T), the prediction error
+of the MLP, and the uncertainties in the reference state
+energy estimate, i.e. e0 in Eq.(1). We verified that, in our
+case, the error produced by the fit is negligible compared
+to the other two sources, which we will discuss next.
+As mentioned before, we followed Ref. 34 to construct
+our MLPs and used a GKR model based on a modified
+version of the SOAP kernel [33].
+Our final dataset,
+including both training and test sets, comprises 871
+configurations selected through an iterative procedure
+with 128 hydrogen atoms each, where we calculated
+energies, pressures and forces at the VMC and LRDMC
+levels. These configurations correspond to temperatures
+from 4 kK up to 35 kK and Wigner-Seitz radii from 1.80
+Bohr to 2.12 Bohr. Finite size corrections have also been
+estimated using the KZK functional [53].
+Details on the training set construction and the QMC
+calculations, together with the performances of all MLP
+models can be found in the SM [45]. In particular we
+found a final root mean square error, calculated on the
+test set, of the order of 20 meV/atom for the energy, 130
+meV/˚A for the forces, and 0.1 GPa for the pressures.
+At this point, it is worth to highlight some favourable
+
+4
+0
+1
+2
+3
+4
+5
+6
+7
+8
+ r (Bohr)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+ g(r)
+T = 4 kK
+T = 6 kK
+T = 7 kK
+T = 8 kK
+T = 10 kK
+T = 15 kK
+T = 35 kK
+3.0
+3.5
+4.0
+4.5
+/
+0
+25
+50
+75
+100
+125
+150
+Pressure (GPa)
+0.92
+0.58 0.43
+0.34
+0.21
+0.12
+0.03
+FIG. 2:
+g(r) for several temperatures and densities close to
+the principal Hugoniot, obtained using the LRDMC model.
+The molecular fraction value, m, is reported in the inset,
+beside each point distributed according to their
+corresponding location in the density-pressure space.
+features of our machine learning approach, especially in
+applications where it is coupled with computationally
+expensive methods such as QMC. They can be itemized
+as follows:
+• transferability: the total energy of the system is
+expressed as a sum of local terms [32], therefore our
+models are capable of making accurate predictions
+on
+configurations
+whose
+size
+has
+never
+been
+encountered in the training set. In particular, our
+MLPs find their applicability to systems with an
+arbitrary number of atoms N.
+• efficiency and accuracy:
+within the ∆-learning
+framework, the machine learning task becomes
+easier.
+Indeed, we obtained very accurate QMC
+potentials, by training models on small datasets
+and, thus, by reducing the amount of calculations
+needed.
+Moreover, since the computational cost
+of the ML inference is negligible compared to the
+baseline DFT calculation, we were able to perform
+QMC-driven MD simulations at the cost of a DFT
+dynamics.
+• overfitting prevention: using a local sparsification
+technique based on the farthest point sampling
+(see SM of Ref. 34), we discarded from each
+configuration a possibly large fraction of the
+corresponding N local environments, preventing
+overfitting and allowing for an increased predictive
+power of the model on unseen data.
+Since
+the computational cost of the predictions scales
+with the size of the training set, this procedure
+drastically improves the efficiency of the final
+model.
+We further validated the accuracy of our MLP models
+by comparing the Hugoniot curve obtained using three
+potentials, independently trained with the same target,
+e.g. VMC, but with different baselines. In particular,
+we found the results to be consistent within an error of
+≲ 1% and ≲ 2% for density and pressure, respectively.
+We now turn to the last source of error we identified,
+i.e.
+the one related to the calculation of e0 and p0.
+To estimate the reference state energy and pressure, we
+followed a procedure similar to Ref. 30. We performed a
+path integral molecular dynamics (PIMD) simulation [54]
+on a system of N = 64 deuterium atoms at a temperature
+T = 22 K and density ρ0 = 0.167 g/cm3 (corresponding
+to the initial conditions reported in Ref. 19), using DFT-
+PBE energy and forces. Details of this simulation are
+reported in the SM [45]. From the PIMD trajectory, we
+extracted 170 configurations and we calculated energies
+and pressures with both DFT-PBE and QMC at VMC
+and LRDMC levels, adding the necessary finite size
+corrections.
+The reference sample was generated by
+extracting atomic positions from one of the 128 beads
+taken at random, belonging to de-correlated snapshots
+of the trajectory. Results for e0 for the various methods
+are reported in Tab. I. The reference state pressure p0 is
+not reported, since it is two orders of magnitude smaller
+than the shocked pressure, and thus irrelevant for the
+Hugoniot determination. Also in this case, we studied
+the effect of varying e0 within its confidence interval on
+the Hugoniot density and pressure. Its variability within
+standard deviation leads to shifts in the final principal
+Hugoniot which fall in the stochastic error range of our
+predictions.
+To summarize, we estimated the MLP prediction error
+to be the most relevant source of uncertainty for the
+Hugoniot, yielding, as discussed before, an error of 1%
+and 2% on the relative density and pressure, respectively,
+reflected on the error bars reported in Fig. 1. Notice that
+our Hugoniot curve is consistent with the experiments
+even after considering the possible uncertainties.
+epot (Ha/atom) e0 (Ha/atom)
+PBE
+-0.58217(2)
+-0.58055(2)
+VMC
+-0.58465(3)
+-0.58303(3)
+LRDMC
+-0.58653(2)
+-0.58491(2)
+TABLE I: Estimated potential (epot) and total (e0)
+energies per atom of the reference state at ρ0 = 0.167
+g/cm3 and T = 22 K for different methods.
+
+5
+Conclusions −
+In conclusion,
+using our recently
+proposed workflow for the construction of MLPs, we have
+been able to run reliable VMC- and LRDMC-based MD
+simulations and study the principal deuterium Hugoniot,
+in a pressure range relevant for experiments.
+The
+accuracy of the MLPs employed here has been extensively
+tested, supporting the validity of our calculations. The
+resulting Hugoniot curve shows an excellent agreement
+with the most recent measures, comparable to the best
+DFT functionals and better than previous QMC results.
+Moreover, within the ∆-learning framework, we have
+also been able to treat FT electrons effects in a QMC-
+MLP, and we have thus managed to perform accurate
+simulations at higher temperatures.
+The efficiency
+of this approach could be further improved, e.g., by
+using cheaper baseline potentials than DFT. Longer
+simulations and larger systems will then be at reach.
+Other many-body methods, even more expensive than
+QMC, can also be used as targets for this type of
+MLPs, since the required size of the dataset is at
+least one order of magnitude smaller compared to other
+ML approaches.
+Finally, our MLPs, and in particular
+those trained on LRDMC datapoints, are promising for
+exploring the hydrogen phase diagram by keeping a high
+level of accuracy across a wide range of thermodynamic
+conditions.
+Data availability −
+The machine learning code used
+in this work is available upon request.
+Additional
+information, such as datasets and detailed results of
+the simulations are available at https://github.com/
+giacomotenti/QMC_hugoniot.
+Acknowledgments. The computations in this work have
+mainly been performed using the Fugaku supercomputer
+provided by RIKEN through the HPCI System Research
+Project (Project ID: hp210038 and hp220060) and
+Marconi100 provided by CINECA through the ISCRA
+project No. HP10BGJH1X and the SISSA three-year
+agreement 2022. K.N. is also grateful for computational
+resources from the facilities of Research Center for
+Advanced Computing Infrastructure at Japan Advanced
+Institute of Science and Technology (JAIST).
+A.T. acknowledges financial support from the MIUR
+Progetti di Ricerca di Rilevante Interesse Nazionale
+(PRIN)
+Bando
+2017
+-
+grant
+2017BZPKSZ.
+K.N.
+acknowledges
+a
+support
+from
+the
+JSPS
+Overseas
+Research Fellowships, that from Grant-in-Aid for Early-
+Career Scientists Grant Number JP21K17752, and that
+from Grant-in-Aid for Scientific Research(C) Grant
+Number JP21K03400.
+This work is supported by the
+European Centre of Excellence in Exascale Computing
+TREX - Targeting Real Chemical Accuracy at the
+Exascale.
+This project has received funding from
+the European Union’s Horizon 2020 - Research and
+Innovation program - under grant agreement no. 952165.
+We dedicate this paper to the memory of Prof. Sandro
+Sorella (SISSA), who tragically passed away during
+this project, remembering him as one of the most
+influential contributors to the quantum Monte Carlo
+community,and in particular for deeply inspiring this
+work with the development of the ab initio QMC code,
+TurboRVB.
+∗ [email protected]
+† [email protected]
+‡ kousuke [email protected]
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+Journal of chemical theory and computation 18, 118
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+(2020).
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+
+Supplemental material: Principal deuterium Hugoniot via Quantum Monte
+Carlo and ∆-Learning
+Giacomo Tenti∗ and Andrea Tirelli†
+International School for Advanced Studies (SISSA),
+Via Bonomea 265, 34136 Trieste, Italy
+Kousuke Nakano‡
+International School for Advanced Studies (SISSA),
+Via Bonomea 265, 34136 Trieste, Italy and
+School of Information Science, JAIST,
+Asahidai 1-1, Nomi, Ishikawa 923-1292, Japan
+Michele Casula
+Institut de Min´eralogie, de Physique des Mat´eriaux et de Cosmochimie (IMPMC),
+Sorbonne Universit´e, CNRS UMR 7590,
+MNHN, 4 Place Jussieu, 75252 Paris, France
+Sandro Sorella
+International School for Advanced Studies (SISSA),
+Via Bonomea 265, 34136 Trieste, Italy and
+Computational Materials Science Research Team,
+RIKEN Center for Computational Science (R-CCS), Kobe, Hyogo 650-0047, Japan
+(Dated: January 10, 2023)
+1
+arXiv:2301.03570v1  [cond-mat.str-el]  9 Jan 2023
+
+I.
+COMPUTATIONAL DETAILS OF QMC CALCULATIONS
+The Variational Monte Carlo (VMC) and lattice regularized diffusion Monte Carlo (LRDMC) [1]
+calculations in this study were performed by TurboRVB package [2]. The package employs a
+many-body WF ansatz Ψ which can be written as the product of two terms, i.e., Ψ = ΦAS × exp J ,
+where the term exp J and ΦAS are conventionally called Jastrow and antisymmetric parts, re-
+spectively. The antisymmetric part is denoted as the Antisymmetrized Geminal Power (AGP)
+that reads:
+ΨAGP (r1, . . . , rN) =
+ˆA
+�
+Φ
+�
+r↑
+1, r↓
+1
+�
+Φ
+�
+r↑
+2, r↓
+2
+�
+· · · Φ
+�
+r↑
+N/2, r↓
+N/2
+��
+, where ˆA is the an-
+tisymmetrization operator, and Φ
+�
+r↑, r↓�
+is called the paring function [3].
+The spatial part
+of the geminal function is expanded over the Gaussian-type atomic orbitals: ΦAGP
+�
+ri, rj
+�
+=
+�
+l,m,a,b f{a,l},{b,m}ψa,l (ri) ψb,m
+�
+r j
+�
+where ψa,l and ψb,m are primitive Gaussian atomic orbitals, their
+indices l and m indicate different orbitals centered on atoms a and b, and i and j are coordi-
+nates of spin up and down electrons, respectively, and f{a,l},{b,m} are the variational parameters. In
+this study, a basis set composed of [4s2p1d] Gaussian atomic orbitals (GTOs) was employed
+for the atomic orbitals of the antisymmetric part.
+The pairing function can be also written
+as ΦAGPn
+�
+ri, r j
+�
+= �M
+k=1 λkφk(ri)φk(rj) with λk > 0, where φk(r) is a molecular orbital, i.e.,
+φk(r) = �L
+i=1 ci,kψi(r). When the paring function is expanded over M molecular orbitals where
+M is equal to half of the total number of electrons (N/2), the AGP coincides with the Slater-
+Determinant ansatz. In this study, we restricted ourselves to a Jastrow-Slater determinant (JSD) by
+setting M = 1
+2 ·N, wherein the coefficients of atomic orbitals, i.e., ci,k, were obtained by the build-in
+Density Functional theory (DFT) package (prep), and were fixed during a VMC optimization.
+The Jastrow term is composed of one-body, two-body and three/four-body factors (J = J1 +
+J2 + J3/4). The one-body and two-body factors are essentially used to fulfill the electron-ion and
+electron-electron cusp conditions, respectively, and the three/four-body factor is employed to con-
+sider further electron-electron correlations (e.g., electron-nucleus-electron). The one-body Jastrow
+is decomposed into the so-called homogeneous and inhomogeneous parts, i.e., J1 = Jhom
+1
++ Jinh
+1 .
+The homogeneous one-body Jastrow factor is J1
+hom (r1, . . . , rN) = �
+i,I
+�
+−(2ZI)3/4u
+�
+2ZI
+1/4 |ri − RI|
+��
+where ri are the electron positions, RI are the atomic positions with corresponding atomic number
+ZI, and u (r) is a short-range function containing a variational parameter b: u (r) = b
+2
+�
+1 − e−r/b�
+.
+The inhomogeneous one-body Jastrow factor Jinh
+1
+is represented as:
+∗ [email protected]
+† [email protected]
+‡ [email protected]
+2
+
+Jinh
+1 (r1, . . . , rN) = �N
+i=1
+�Natom
+a=1
+��
+l Ma,lχa,l (ri)
+�
+, where ri are the electron positions, Ra are the
+atomic positions with corresponding atomic number Za, l runs over atomic orbitals χa,l (e.g., GTO)
+centered on the atom a, Natom is the total number of atoms in a system, and {Ma,l} are variational
+parameters. The two-body Jastrow factor is defined as: J2 (r1, . . . rN) = exp
+��
+i<j v
+����ri − r j
+���
+��
+,
+where v (r) = 1
+2r · (1 − F · r)−1 and F is a variational parameter. The three-body Jastrow factor
+is: J3/4 (r1, . . . rN) = exp
+��
+i< j ΦJas
+�
+ri, r j
+��
+, and ΦJas
+�
+ri, rj
+�
+= �
+l,m,a,b ga,l,m,bχJas
+a,l (ri) χJas
+b,m
+�
+r j
+�
+, where the
+indices l and m again indicate different orbitals centered on corresponding atoms a and b. In this
+study, the coefficients of the three/four-body Jastrow factor were set to zero for a � b because it
+significantly decreases the number of variational parameters while rarely affects variational ener-
+gies. A basis set composed of [3s] GTOs was employed for the atomic orbitals of the Jastrow part.
+The variational parameters in the Jastrow factor were optimized by the so-called stochastic recon-
+figuration [4] implemented in TurboRVB. Total energies and forces are calculated at the VMC and
+the LRDMC levels with the optimized wavefunctions. The LRDMC calculations were performed
+by the original single-grid scheme [1] with the discretization grid size a = 0.20 Bohr. To alleviate
+the one-body finite-size effects, we have used twisted average boundary conditions (TABC) with
+a 4 × 4 × 4 Monkhorst-Pack grid.
+To obtain a statistically meaningful value of VMC and LRDMC forces with finite variance [5],
+the so-called reweighting techniques are needed because the Hellmann–Feynman (HF) and Pulay
+terms may diverge when the minimum electron–nucleus distance vanishes and when an electronic
+configuration is close to the nodal surface, respectively [6]. The infinite variance of the first term
+is cured by applying the so-called space-warp coordinate transformation (SWCT) algorithm [6–9],
+whereas that of the second term can be alleviated by modifying the VMC sampling distribution
+using a modified trial wave function that differs from the original trial wave function only in the
+vicinity of the nodal surface [10], which we dub the Attaccalite and Sorella (AS) regularization.
+The AS regularization is not an optimal regularization for this purpose because it enforces a finite
+density of walkers on the nodal surface [11]. Therefore, in this study, we employed the regular-
+ization technique recently proposed by Pathak and Wagner [12] combined with mixed-averaged
+forces proposed by Reynolds [13].
+3
+
+II.
+MLP TRAINING AND VALIDATION
+A.
+Dataset construction
+To construct our dataset, we performed a first set of PBE MD simulations on a system of N =
+128 atoms for temperatures in the range [4kK, 20kK] and densities in the range [1.80 Bohr, 2.20
+Bohr], from which we extracted 500 decorrelated snapshots. We then added other configurations
+according to an active learning scheme: with a model trained using this first dataset we ran MD
+simulations and iteratively selected new points where the MLP performances were expected to be
+poor. In particular we did this by monitoring, for each unseen configuration, the quantity
+χ = 1
+N
+N
+�
+i=1
+min
+µ∈training set K(Ri, Rµ)
+(1)
+where K(Ri, Rµ) is the normalized SOAP kernel between the i-th local environment of the con-
+figuration Ri and the µ-th local environment in the training set Rµ. The number χ defined in (1)
+gives a quantitative measure of ”how far” the unknown configuration is from what is already in-
+cluded in the training set. At the end the final dataset, i.e., the one for which χ did not drop under
+a certain fixed threshold (0.80 in our case) during the dynamics, comprised 871 configurations
+of 128 atoms in total. The final range of temperatures and Wigner-Seitz radii spanned by these
+configurations was [4 kK : 35 kK ] and [1.80 Bohr : 2.12 Bohr ], respectively.
+B.
+Training details
+For the training procedure we followed the strategy outlined in [14, §I.B]: the cost function C
+employed is the regularised weighted sum of the RMSE on the observables, i.e.
+C = C(cµ) = αMSE(E, ˆE(cµ)) + βMSE(F, ˆF(cµ)) + γMSE(P, ˆP(cµ)) + λ||cµ||2,
+where E, F, P are the vectors representing the observables obtained through QMC simulations and
+ˆE, ˆF, ˆP are the observables computed through GKR. For the choice of the model hyperparameters,
+a cross-validation test led the following:
+• the cutoff radius used to compute local environments has been set to rc = 5.0 Bohr.
+• the parameters α, β, γ and λ determining the cost function C(cµ) have been set to 10−1, 1, 102
+and 10−5 respectively.
+4
+
+C.
+Models performance
+The performances of the models employed are measured through the root mean squared error
+(RMSE) on the observables on which the models were trained. Such RMSEs are reported in Table
+I.
+D.
+Effect of different baselines
+In order to further validate the accuracy of a MLP, a common strategy is to compare the results
+of the dynamics obtained using the trained model with those obtained with the target ab initio
+method directly, at least for some small system sizes. In our case this is not an easy task, given the
+large computational time that would be needed for computing energies and forces at each step with
+QMC. An alternative way to establish the performances of the models is to look at the variance of
+the results obtained with MLPs trained using different baselines. The Hugoniot function H(ρ, T)
+and pressure at T = 8 kK are shown in Fig.(1).
+We can estimate the error produced by using different baselines as 1% in the Hugoniot density
+and 1 − 2% (≲ 1GPa) in the Hugoniot pressure.
+III.
+REFERENCE STATE CALCULATIONS
+As explained in the main text, a crucial part in the numerical determination of the Hugoniot is
+to estimate the reference state energy per atom e0 and pressure p0. In particular, having a precise
+Nconf Nenvs RMSEE (Ha / atom) RMSE f (Ha / Bohr) RMSEp (a.u.) RMSEp (GPa)
+VMC - PBE
+666 4965
+8.34 × 10−4
+2.396 × 10−3
+2.09 × 10−6
+0.061
+VMC - LDA
+778 4966
+8.25 × 10−4
+3.358 × 10−3
+2.48 × 10−6
+0.073
+VMC - DF1
+785 4961
+7.20 × 10−4
+2.215 × 10−3
+1.63 × 10−6
+0.048
+LRDMC - PBE
+666 4965
+7.28 × 10−4
+2.507 × 10−3
+3.36 × 10−6
+0.098
+LRDMC - LDA 666 4965
+8.44 × 10−4
+3.374 × 10−3
+3.64 × 10−6
+0.11
+TABLE I: Training set size and value of the RMSE on different observables as calculated on the
+test set for the final models used in the simulations.
+5
+
+1.90
+1.95
+2.00
+rs
+0.02
+0.00
+H(rs,T)
+T = 8000K
+PBE + ML
+LDA + ML
+DF1 + ML
+(a)
+1.90
+1.95
+2.00
+rs
+40
+50
+Pressure( GPa )
+T = 8000K
+PBE + ML
+LDA + ML
+DF1 + ML
+(b)
+FIG. 1: Results for the Hugoniot function H(rs, T) for T = 8 kK with different MLPs trained on
+VMC data, using different baselines potentials.
+value of e0 within the target method is important to take advantage of possible error cancellation
+effects and remove biases related to finite basis sets. We considered a system of N = 64 deuterium
+atoms at T0 = 22K and ρ0 = 0.167g/cm−3 and ran a path integral Ornstein-Uhlenbeck molecular
+dynamics [15] (PIOUMD) simulation to account for quantum effects, which are required because
+of the light deuterium mass and low temperature. Forces and energy were calculated with Density
+functional theory (DFT) through the Quantum-Espresso package. We checked the dependence of
+thermodynamic quantities on the number of replicas (or beads) M and on the choice of the DFT
+functional by studying the quantum kinetic energy T for several values of M using the BLYP and
+PBE functionals. In particular we considered two estimators for T, namely the virial and primitive
+(or Barker) estimator, given respectively by
+TM,vir = N
+2β + 1
+2M
+3N
+�
+i=1
+M
+�
+j=1
+�
+x(j)
+i
+− ¯xi
+�
+∂x(j)
+i V
+(2)
+TM,pri = 3NM
+2β
+− mM
+2β2ℏ2
+M
+�
+j=1
+�
+x( j)
+i
+− x(j−1)
+i
+�2
+(3)
+where M is the number of replicas used in the PIOUMD simulation, x( j) =
+�
+x(j)
+1 , . . . , x( j)
+3N
+�
+are
+the coordinates of the system belonging to the j-th bead, ¯xi =
+1
+M
+�M
+j=1 x(j)
+i
+is the centroid position
+and β = kBT0. The results are shown in Fig.(2).
+We noticed that a very large number of replicas is necessary for having a sufficiently converged
+result, while the value obtained with the two functionals is extremely similar for all values of M.
+6
+
+50
+100
+150
+200
+250
+# beads
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+Kinetic energy (Ha) 
+virial PBE
+primitive PBE
+virial BLYP
+primitive BLYP
+FIG. 2: Convergence of virial and primitive estimators for the quantum kinetic energy, as
+computed with Eqs. (2) (3), with the number of replica used in the PIMD simulation.
+At the end we chose to use the PBE functional and M = 128 replicas to have a reasonable trade-
+off between convergence and computational cost. For the DFT calculation we used a 60 Ry plane
+waves cutoff and a 2 × 2 × 2 Monkhorst-Pack k point mesh; for the dynamics we used a time
+step of 0.3 fm and let the system thermalize for 0.3 ps. We then extracted one configuration from
+a randomly chosen bead every 10 MD steps, for a total of Nsample = 170 snapshots. Finally the
+potential energy of these configurations was calculated using the appropriate method (PBE, VMC
+or LRDMC). We then estimated e0 for each method as
+e0 = 1
+N
+��������
+1
+Nsample
+�
+sample
+Epot (xi) + T PBE
+256,pri
+��������
+(4)
+using the value of the primitive estimator at M = 256 beads as the best guess for the converged
+value of the kinetic energy. The approximation for the potential energy was checked by running
+PBE simulations on this set and confirming that the ”true” mean value (as calculated by averaging
+over the beads and the trajectory) was consistent with our estimate obtained by averaging over the
+sample.
+7
+
+IV.
+FINITE SIZE CORRECTIONS
+In this section we investigate the effect of finite size corrections (as estimated using the KZK
+functional [16]) on our results. In Fig.(3) we show the Hugoniot function ( at T = 4 kK and
+T = 8 kK) given by two models trained on VMC and VMC with finite size corrections respectively,
+both with a PBE baseline potential. The difference between the two turns out to be similar to the
+prediction error evaluated in Sec. II D, for the system size used in the simulations (i.e., N = 128).
+At the end we chose to apply finite size correction only for the model trained with LRDMC data.
+2.20
+2.25
+rs (a.u.)
+0.0025
+0.0000
+0.0025
+0.0050
+Hugoniot (Ha)
+ T = 4000K
+VMC
+VMC- FSC
+(a)
+1.90
+1.95
+2.00
+rs (a.u.)
+0.01
+0.00
+Hugoniot (Ha)
+ T = 8000K
+VMC
+VMC- FSC
+(b)
+FIG. 3: Hugoniot function obtained using two MLPs trained on the difference between PBE and
+VMC with and without finite size corrections respectively, for T = 4 kK and T = 8 kK.
+V.
+FINITE TEMPERATURE DFT SIMULATIONS
+Using Mermin’s extension of the Hohenberg and Kohn theorems to non-zero temperature [17]
+we can treat finite temperature electrons in DFT by appropriately occupying the bands of the
+system according to the Fermi-Dirac distribution and minimizing the Helmholtz free energy func-
+tional A = E − TS . In this work we performed finite temperature DFT (FT-DFT) simulations to
+obtain the PBE Hugoniot and estimating the effect on the QMC Hugoniot. In the former case we
+used the zero temperature PBE functional for the simulations. Even if this is not rigorous, recent
+FT-DFT results on the Hugoniot using a temperature dependent GGA functional [18] have shown
+that for T ≲ 40 kK this approximation provides consistent results. For the latter application, we
+decided to use an explicitly temperature dependent functional to replace the LDA baseline of one
+8
+
+of the MLPs. In particular we used the corr-KSDT functional [19, 20], as implemented in the
+Libxc [21] library. This functional has the nice property to recover the standard PZ-LDA func-
+tional (that was used for the construction of the MLP under consideration) when T = 0K. In Fig.
+(4) we show the convergence of the free energy and some force components with the number of
+bands calculated for the KSDT functional at two values of temperature. In the simulations we
+decided to use 120 bands for T = 10 kK and T = 15 kK and 150 bands for T = 35 kK.
+100
+200
+300
+# of Bands
+0.5484
+0.5483
+0.5482
+0.5481
+Aft (Ha / atom)
+T= 8kK
+(a)
+100
+200
+300
+# of Bands
+0.025
+0.000
+0.025
+0.050
+Force (Ha/bohr)
+T= 8kK
+(b)
+100
+200
+300
+# of Bands
+0.580
+0.575
+0.570
+0.565
+Aft (Ha / atom)
+T= 30kK
+(c)
+100
+200
+300
+# of Bands
+0.05
+0.00
+0.05
+Force (Ha/bohr)
+T= 30kK
+(d)
+FIG. 4: Convergence of the free Energy A = E − TS and some forces components vs the number
+of bands calculated in the DFT for T = 8 kK (4a, 4b) and T = 30 kK (4c, 4d). Bands are
+occupied using the Fermi-Dirac distribution at the appropriate temperature.
+9
+
+VI.
+COMPARISON WITH QMC CALCULATIONS OF REF. [22].
+The equations of state at T = 8 kK reported in Ref. [22] for both variational and reptation
+Monte Carlo are shown in Fig. (5), together with the VMC-MLP, LRDMC-MLP and the ab initio
+PBE ones. From this figure we can observe a huge discrepancy between the pressure estimated
+with our MLPs and the one in Ref. [22], which causes a sizable difference in the position of the
+Hugoniot. In our case the VMC and LRDMC pressures, which we then used for training, were
+calculated using the adjoint algorithmic differentiation method to obtain directly the derivative of
+the total energy with respect to the cell parameters. For LDA orbitals, as the ones used in this
+work, the pressure obtained with this procedure is not biased (see [14, §I.A]). Instead, in Ref. [22]
+a virial estimator was used, which can in principle produce discrepancies of the order of magnitude
+observed here, as shown in Ref. [23]. We also point out that the similarity between the LRDMC
+and VMC results suggests an overall robustness of our pressure estimation.
+1.80
+1.85
+1.90
+1.95
+2.00
+2.05
+rs (a.u.)
+30
+35
+40
+45
+50
+55
+Pressure (GPa)
+PBE
+LRDMC
+VMC
+VMC Ref. 22
+RMC Ref. 22
+FIG. 5: Average pressure vs rs for T = 8 kK. Results obtained with our MLP are shown together
+with the ones reported in [22]
+10
+
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+Y. Luo, G. Mazzola, A. Zen, and S. Sorella, Turborvb: A many-body toolkit for ab initio electronic
+simulations by quantum monte carlo, J. Chem. Phys. 152, 204121 (2020).
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+atoms, J. Chem. Phys. 119, 6500 (2003).
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+waals attraction by quantum monte carlo methods, J. Chem. Phys. 127, 014105 (2007).
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+carlo: Application to phonon dispersion calculations, Phys. Rev. B 103, L121110 (2021).
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+calculations in quantum monte carlo, The Journal of Chemical Physics 156, 034101 (2022).
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+12
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+STAR-RIS Assisted Over-the-Air Vertical Federated
+Learning in Multi-Cell Wireless Networks
+Xiangyu Zeng∗†‡, Yijie Mao∗, and Yuanming Shi∗
+∗School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China
+†Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, China
+‡University of Chinese Academy of Sciences, Beijing 100049, China
+E-mail: {zengxy, maoyj, shiym}@shanghaitech.edu.cn
+Abstract—Vertical federated learning (FL) is a critical enabler
+for distributed artificial intelligence services in the emerging
+6G era, as it allows for secure and efficient collaboration of
+machine learning among a wide range of Internet of Things
+devices. However, current studies of wireless FL typically con-
+sider a single task in a single-cell wireless network, ignoring
+the impact of inter-cell interference on learning performance.
+In this paper, we investigate a simultaneous transmitting and
+reflecting reconfigurable intelligent surface (STAR-RIS) assisted
+over-the-air computation based vertical FL system in multi-cell
+networks, in which a STAR-RIS is deployed at the cell edge
+to facilitate the completion of different FL tasks in different
+cells. We establish the convergence of the proposed system
+through theoretical analysis and introduce the Pareto boundary
+of the optimality gaps to characterize the trade-off among cells.
+Based on the analysis, we then jointly design the transmit and
+receive beamforming as well as the STAR-RIS transmission and
+reflection coefficient matrices to minimize the sum of the gaps of
+all cells. To solve the non-convex resource allocation problem, we
+introduce a successive convex approximation based algorithm.
+Numerical experiments demonstrate that compared with con-
+ventional approaches, the proposed STAR-RIS assisted vertical
+FL model and the cooperative resource allocation algorithm
+achieve much lower mean-squared error for both uplink and
+downlink transmission in multi-cell wireless networks, resulting
+in improved learning performance for vertical FL.
+I. INTRODUCTION
+Federated learning (FL) is a machine learning (ML) ap-
+proach that enables multiple parties to collaboratively train a
+learning model without revealing their individual data. This
+is beneficial in a variety of fields where data privacy is a
+concern, as FL allows parties to maintain control over their
+own data while still benefiting from the combined knowledge
+of all parties. In modern wireless Internet of Things (IoT)
+networks, data is often collected from various types of devices
+[1]. To facilitate data analysis in such settings, vertical FL,
+a variation of FL that is designed to address the challenges
+of training machine learning models on vertically partitioned
+data silos, is commonly adopted [2]–[6].
+One major issue that prevents the implementation of (ver-
+tical) FL in real-world application is the communication
+latency. To address this issue, over-the-air computation (Air-
+Comp) has been proposed to facilitate fast wireless data
+aggregation. By utilizing the superposition property of wire-
+less multiple access channels (MAC) to concurrently transmit
+and aggregate local updates, AirComp significantly reduces
+communication latency compared to orthogonal transmission.
+Previous research has explored the use of AirComp in FL,
+such as the joint design of device selection and beamforming
+for fast global model aggregation in [5], and the development
+of a broadband analog aggregation scheme for low latency FL
+with linear growth of latency reduction ratio in [7].
+On the other hand, the coexistence of multiple FL tasks
+in multi-cell networks has yet to be fully explored. Though
+the authors in [8] have studied the bandwidth allocation
+for multiple FL tasks, the system model is limited to a
+single-cell network and the impact of inter-cell interference
+on FL performance remains unplumbed. It has been well
+investigated that reconfigurable intelligent surface (RIS), a
+metasurface composed of reconfigurable passive elements,
+can modify the propagation environment of wireless signal
+and reduce multi-cell interference [9]. However, conventional
+RISs are reflecting only with limited wireless coverage [10].
+The recently introduced simultaneous transmitting and reflect-
+ing RIS (STAR-RIS), which allows the source and destination
+to be located at either side of the metasurface, has been
+recognized as a promising strategy to enhance the coverage
+of each cell and further reduce inter-cell interference [11].
+STAR-RIS is therefore a promising technique to facilitate FL
+in multi-cell networks. To the best of our knowledge, STAR-
+RIS assisted vertical FL has not been studied yet.
+In this paper, inspired by the benefits of AirComp for global
+aggregation [12] and the merits of STAR-RIS in multi-cell
+networks, we fill the research gap and propose a STAR-RIS
+assisted AirComp-based vertical FL in multi-cell networks,
+where a STAR-RIS is deployed at the cell edge to assist each
+cell in completing different FL tasks. Through theoretical
+analysis, we demonstrate the convergence of our proposed
+vertical FL process and introduce the Pareto boundary of the
+gap region to characterize the trade-off performance among
+multiple cells. This allows us to formulate an optimization
+problem with the aim of minimizing the sum of error-induced
+gaps for all cells using the proposed algorithm based on suc-
+cessive convex approximation (SCA). Numerical experiments
+confirm the validity of our theoretical analysis and show the
+superiority of our proposed approach.
+II. SYSTEM MODEL
+A. Learning Framework
+Consider a STAR-RIS assisted multi-cell wireless net-
+work consisting of M base stations (BS) with N an-
+arXiv:2301.05545v1  [cs.IT]  13 Jan 2023
+
+tennas, where BS m
+∈
+M
+=
+{1, 2, . . . , M} aims
+to
+train
+an
+ML
+model
+by
+coordinating
+Km
+single-
+antenna devices located in cell m. Specifically, device
+k ∈ Km =
+��m−1
+l=1 Kl + 1, �m−1
+l=1 Kl + 2, . . . , �m−1
+l=1 Kl+
+Km} is associated with BS m. And there is one STAR-RIS
+equipped with Q passive reflecting/transmitting elements, de-
+ployed at the cell-edge of all cells to boost the signal strength
+of edge devices. Each cell is equipped with a vertically
+partitioned dataset, where different devices hold different
+features of the same samples. For simplicity, we assume that
+each cell has the same number of samples and that devices
+within each cell contain the same number of non-overlapping
+features. Let Dm = {(xi
+m,1, · · · , xi
+m,Km), yi
+m}Lm
+i=1 denote the
+whole training dataset of Lm samples in cell m, where xi
+m,k
+denotes the partial features of sample i located at device k in
+cell m, and yi
+m denotes the corresponding label. In vertical
+FL, it is assumed that the BS holds all labels ym = {yi
+m}Lm
+i=1,
+and device k is only available to its own local feature set
+Dm,k = {xi
+m,k}Lm
+i=1. And xi
+m = [(xi
+m,1)T, · · · , (xi
+m,Km)T]T
+denotes the overall feature vector of sample i.
+The goal of vertical FL in cell m is to collaboratively learn
+a global model wm (concatenated vector of wk for k ∈ Km)
+that maps an input to the corresponding prediction through
+a continuously differentiable function σ(·). Since features of
+one sample are distributed at different devices, we assume
+that device k maps the local feature xk to local prediction
+result gk(wk; xk). This paper considers a linear form for the
+local prediction function, i.e., gk(wk; xk) = wT
+k xk. By ag-
+gregating local prediction results, the final prediction in cell m
+can be obtained by σ(wm; xm) = σ(�
+k∈Km gk(wk; xk)) =
+σ(wT
+mxm). In order to learn the global model wm in cell m,
+we propose to minimize the loss function as
+min
+wm F(wm) =
+1
+Lm
+Lm
+�
+i=1
+f
+�
+σ(wT
+mxi
+m); yi
+m
+�
+,
+(1)
+where f(·) is the sample-wise loss function.
+In our multi-cell system, each cell performs a unique FL
+task using the full batch gradient descent (GD) approach,
+which is described in the following subsection. We assume
+universal frequency reuse, meaning that all cells share the
+same frequency channel, leading to inter-cell interference.
+B. GD Algorithm for Vertical FL
+In this subsection, we introduce the framework of GD
+algorithm for vertical FL. For brevity, the subscript of cell m
+is omitted for Lm, wm, xm, ym. The GD algorithm specified
+in this subsection is applied for all cells. Let ∇F(w) denote
+the gradient of F respect to w, which is calculated as
+∇F(w) = 1
+L
+L
+�
+i=1
+∇f(σ(wTxi); yi),
+(2)
+where
+∇f(σ(wTxi); yi)
+denote
+the
+gradient
+of
+f(σ(wTxi); yi) respect to w. Based on the chain rule,
+the gradient of f is rewritten as
+∇f(σ(wTxi); yi) = G(wTxi; yi)xi,
+(3)
+where G(wTxi; yi) = ∂f(σ(wTxi); yi)/∂wTxi is an auxil-
+iary function. Hence, ∇F(w) can be rewritten as
+∇F(w) = 1
+L
+L
+�
+i=1
+G(wTxi; yi)xi.
+(4)
+Recall that the BS holds all labels y, so G(wTxi; yi) can
+be calculated at the BS only if the BS can access the
+aggregation of local predictions {wTxi}L
+i=1. Specifically, at
+the t-th communication round, the BS and the edge devices
+in each cell perform the following three procedures:
+Broadcasting: The BS computes {G((w(t))Txi; yi)}L
+i=1
+and broadcasts the result back to its corresponding devices.
+Local model update: After broadcasting, device k com-
+putes the partial gradient ∇kF(wk) with local data Dk, given
+as
+∇kF(wk) = 1
+L
+L
+�
+i=1
+G(wTxi; yi)xi
+k.
+(5)
+Each device can thus update its local model by taking a step
+of GD with learning rate µ(t) as
+w(t+1)
+k
+= w(t)
+k
+− µ(t)∇kF(w(t)
+k ),
+(6)
+where w(t)
+k
+is the local model of device k at the t-th round.
+Local prediction and global aggregation: device k com-
+putes the local prediction results {(w(t+1)
+k
+)Txi
+k}L
+i=1 and
+sends to the BS. And BS aggregates them to get final
+prediction result {(w(t+1))Txi}L
+i=1.
+Since the BS only needs the aggregation of local prediction
+results, i.e., neither local features nor local models need be
+uploaded to the BS, which significantly enhances privacy pro-
+tection. In addition, the communication efficiency is improved
+since the local prediction result is usually low-dimensional.
+C. Communication Model
+In this subsection, the proposed communication model is
+delineated with a special focus on the STAR-RIS assisted
+uplink and downlink transmission models.
+1) STAR-RIS: The STAR-RIS is a type of RIS that can
+produce omnidirectional radiation by implementing equiva-
+lent electric and magnetic currents in its hardware. It has
+three protocols for use in wireless networks: energy splitting,
+mode switching, and time switching. In this article, we focus
+on the mode-switching protocol, in which each element of
+the STAR-RIS can operate in either the reflection mode (R
+mode) or the transmission mode (T mode). Such on-off type
+of operating protocol is simpler to implement compared to
+the energy splitting protocol. Specifically, one group consists
+of Qt elements operating in the T mode, while the other
+group contains Qr elements operating in the R mode, where
+Qt + Qr = Q. Accordingly, the STAR-RIS transmission-
+coefficient
+and
+reflection-coefficient
+matrices
+are
+given
+by Θt
+= diag
+��
+βt
+1ejθt
+1,
+�
+βt
+2ejθt
+2, . . . ,
+�
+βt
+Qejθt
+Q
+�
+and
+Θr
+= diag
+��
+βr
+1ejθr
+1,
+�
+βr
+2ejθr
+2, . . . ,
+�
+βr
+Qejθr
+Q
+�
+, respec-
+tively, where βt
+q, βr
+q ∈ {0, 1}, βt
+q + βr
+q = 1, and θt
+q, θr
+q ∈
+
+[0, 2π), ∀q ∈ {1, 2, . . . , Q}. The M cells can be divided
+into two groups Mr and Mt. Specifically, cell m is in the
+reflection dimension with m ∈ Mr and in the transmission
+dimension with m ∈ Mt.
+Let hm,k ∈ CN, hr
+k ∈ CQ and Gm ∈ CQ×N denote the
+equivalent channels from edge device k to BS m, from edge
+device k to the STAR-RIS, and from the STAR-RIS to BS
+m, respectively. The combined channel from the k-th edge
+device to the BS m via the STAR-RIS can be written as
+¯hm,k =
+� hm,k + GH
+mΘthr
+k, ∀m ∈ Mt,
+hm,k + GH
+mΘrhr
+k, ∀m ∈ Mr.
+Note that the uplink and downlink STAR-RIS matrices can be
+separatively designed. For simplify, we write Θt and Θr for
+uplink and downlink transmission in terms of Θul and Θdl,
+respectively.
+2) Uplink transmission: In the uplink transmission, we
+assume the devices communicate with the BS via AirComp,
+which has a wide range of FL applications.
+Specifically, we denote sk = [s1
+k, s2
+k, · · · , sL
+k ]T ∈ CL
+as the local prediction results at device k, where the local
+prediction result of the i-th sample si
+k = wT
+k xi
+k. At each
+time slot i ∈ {1, 2, · · · , Lm}, each device in cell m sends
+the corresponding prediction result of the i-th sample to BS
+m. And we assume that sk is normalized with zero mean
+and unit variance [13]. We denote gm(i) = �
+k∈Km si
+k as the
+target function to be estimated through AirComp at the i-th
+time slot. To simplify the notation, we omit the time index by
+writing g(i) and si
+k as g and sul
+k , respectively. And we assume
+that the signals transmitted by all devices are synchronized
+at the BS. Then the received signal at BS m is given by
+yul
+m =
+�
+k
+¯hm,kbksul
+k + nul
+m,
+(7)
+where bk ∈ C is the transmit scalar at device k, and nul
+m is the
+additive white Gaussian noise with zero mean and variance
+(σul)2 at BS m. The transmit power constraint at device k is
+E(|bksul
+k |2) = |bk|2 ≤ P ul, where P ul > 0 is the maximum
+transmit power. The scaled signal received at BS m is
+¯gm =
+1
+√ηm
+rH
+myul
+m =
+1
+√ηm
+rH
+m
+�
+k∈K
+¯hm,kbksul
+k + rH
+mnul
+m
+√ηm
+, (8)
+where rm ∈ CN is the receive beamforming vector and ηm
+is a normalizing factor for cell m. To compensate for the
+phase distortion introduced by complex channel responses,
+the transmit scalar at device k in cell m is set to bk =
+√ηm
+(rH
+m¯hm,k)H
+|rHm¯hm,k|2 , ∀k ∈ Km, and ηm can be expressed as
+ηm = P ul mink∈Km |rH
+m¯hm,k|2. Then the estimated function
+at BS for cell m is given as
+ˆgm = ℜ{¯gm}
+= ℜ{gm +
+1
+√ηm
+rH
+m
+�
+l̸=m
+�
+j∈Kl
+¯hm,jbjsul
+j + rH
+mnul
+m
+√ηm
+�
+��
+�
+eulm
+}
+= gm + ℜ{eul
+m}
+(9)
+3) Downlink transmission: After obtaining the estimate
+ˆgm in the cell m, BS m computes G(ˆgm; y) with noisy
+aggregation ˆgm, and then broadcasts the result to the associ-
+ated devices in Km. And we write G(ˆgm; y) in terms of Gm
+for simplify. Without loss of generality, we assume that the
+transmitted signal follows the standard Gaussian distribution,
+i.e., Gm ∼ CN(0, 1). The received signal at device k is
+ydl
+k =
+�
+m
+¯hH
+m,ktmGm + ndl
+k ,
+(10)
+where tm denotes the transmit beamforming vector at BS
+m, and ndl
+k ∼ CN
+�
+0, (σdl)2�
+is the additive white Gaus-
+sian noise with zero mean and variance (σdl)2 at device
+k. The maximum transmit power at BS m is P dl, i.e.,
+E(∥Gmtm∥2) = ∥tm∥2 ≤ P dl.
+To compensate for the phase distortion introduced by
+complex channel responses, the receive scalar at device k
+in cell m is set to rk =
+(¯hH
+m,ktm)H
+|¯hH
+m,ktm|
+2 . The estimated Gm at
+device k is given as
+ˆGm,k = ℜ{rkydl
+k }
+= ℜ{Gm +
+(¯hH
+m,ktm)H
+���¯hH
+m,ktm
+���
+2
+�
+��
+l̸=m
+¯hH
+l,ktlGl + ndl
+k
+�
+�
+�
+��
+�
+¯edl
+k
+}
+= Gm + ℜ{¯edl
+k }.
+(11)
+Note that the uplink noise is embedded in function Gm. In
+order to directly describe the effective noise, we expand Gm
+to its first-order Taylor expansion as follows
+ˆGm,k
+= Gm + ℜ{¯edl
+k }
+= G(gm + ℜ{eul
+m}; y) + ℜ{¯edl
+k }
+= G(gm; y) + G
+′(gm; y)ℜ{eul
+m} + O(|ℜ{eul
+m}|2) + ℜ{¯edl
+k }
+≈ G(gm; y) + G
+′(gm; y)ℜ{eul
+m} + ℜ{¯edl
+k }
+�
+��
+�
+edl
+k
+,
+(12)
+where G′(·) is the first derivative of G(·). Assume that the
+noise amplitude is small, the term O(|ℜ{eul
+m}|2) is neglected,
+which implies the last approximation in (12).
+III. CONVERGENCE ANALYSIS AND PROBLEM
+FORMULATION
+A. Convergence Analysis
+In previous work [12], [14], [15], the convergence analysis
+of the AirComp-based vertical FL process in each cell has
+been established under the following assumptions.
+Assumption 1 (α-strongly convexity). The function F(·) is
+assumed to be α-strongly convex on Rd with constant α,
+namely, for all x, y ∈ Rd, we have
+F(y) ≥ F(x) + ∇F(x)T(y − x) + α
+2 ∥y − x∥2
+2.
+
+Assumption 2 (β-smoothness). The function F(·) is assumed
+to be β-smooth on Rd with constant β, namely, for all x, y ∈
+Rd, we have
+F(y) ≤ F(x) + ∇F(x)T(y − x) + β
+2 ∥y − x∥2
+2.
+Theorem 1 (Convergence of vertical FL process). Suppose
+that Assumption 1 and 2 hold, setting the learning rate to
+be 0 < µ(t) ≤
+1
+β , then the expected optimality gap after T
+communication rounds is upper bounded by
+E
+�
+F(w(T )
+m ) − F(w∗
+m)
+�
+≤ ρT E
+�
+F(w(0)
+m ) − F(w∗
+m)
+�
++
+1
+2βL2
+T −1
+�
+t=0
+ρT −t−1 �
+k∈Km
+�
+Φ1,kE[|ℜ{eul
+m}|2] + Φ2,kE[|ℜ{¯edl
+k }|2]
+�
+,
+(13)
+where ρ = 1−α/β, Φ1,k = �L
+i=1 ∥(Gi
+m,k)
+′xi
+k∥2
+2 and Φ2,k =
+�L
+i=1 ∥xi
+k∥2
+2.
+Proof. Please refer to previous work [12].
+B. Problem Formulation
+According to Theorem 1, the convergence optimality gap
+is largely determined by the mean-squared-error (MSE) of
+both gm and Gm. However, solely optimizing MSE for each
+cell through AirComp may result in significant inter-cell
+interference in the considered multi-cell wireless networks,
+which can negatively impact the learning performance of
+other cells. As such, it is necessary to carefully balance the
+learning performance among various FL tasks in multiple
+cells through a cooperative design.
+We begin by identifying the gap region G, to be the set of
+tuples (∆1, ∆2, . . . , ∆M), which represents the instantaneous
+errors that cause gaps in all cells, and can be achieved simul-
+taneously under specific downlink and uplink transmission
+power constraints. The gap region G can be represented as
+G =
+�
+{(∆1, ∆2, . . . , ∆M)|∆m ≥ Gapm, ∀m ∈ M}, (14)
+where
+Gapm =
+�
+k∈Km
+�
+Φ1,kE[|ℜ{eul
+m}|2] + Φ2,kE[|ℜ{¯edl
+k }|2]
+�
+,
+(15)
+E[|ℜ{eul
+m}|2] =
+�
+l̸=m,j∈Kl
+ηl|rH
+m¯hm,j|2
+ηm|rH
+l ¯hl,j|2 + ∥rm∥2σ2
+ul
+ηm
+,
+E[|ℜ{¯edl
+k }|2] =
+�
+l̸=m |¯hH
+l,ktl|2 + (σdl)2
+���¯hH
+m,ktm
+���
+2
+.
+(16)
+As previously stated, in order to decrease the error-induced
+gap in one cell, the gaps of other cells maybe increased.
+In light of this, our objective is to find a suitable solution
+that allows us to achieve the Pareto boundary of the gap
+region G, so as to balance the performance of learning among
+multiple cells. In this context, the Pareto optimality of a tuple
+is described as follows [16].
+Here, we leverage the profiling technique [17] to char-
+acterize the Pareto boundary by coordinating all BSs to
+minimize the sum of Gap of all cells. Specifically, let
+κ = [κ1, κ2, . . . , κM] denote a given profiling vector, which
+satisfies κm ≥ 0, ∀m ∈ M, and �
+m∈M κm = 1. The gap
+tuple on Pareto boundary can be obtained by solving the
+following problem
+minimize
+ζ,{rm},{tm},Θt,Θr
+ζ
+(17a)
+s.t.
+Gapm ≤ κmζ, ∀m ∈ M
+(17b)
+ζ ≥ 0,
+(17c)
+where ζ denotes the sum of the gaps of all cells. Thus,
+the gap tuple can be represented as (∆1, ∆2, . . . , ∆M) =
+(κ1ζ, κ2ζ, . . . , κMζ), where a smaller value of κm implies a
+more stringent requirement for the gap of cell m.
+Denote ζ = ζul +ζdl, where ζul and ζdl are used to quan-
+tify the sum of instantaneous error-induced gaps generated by
+uplink and downlink transmissions, respectively. Hence, we
+rewrite problem (17) as
+minimize
+ζul,ζdl,{rm},{tm},Θt,Θr
+ζul + ζdl
+(18a)
+s.t.
+Gapul
+m ≤ κmζul, ∀m ∈ M
+(18b)
+Gapdl
+m ≤ κmζdl, ∀m ∈ M
+(18c)
+ζul ≥ 0
+(18d)
+ζdl ≥ 0.
+(18e)
+The downlink and uplink transmissions can be decoupled in
+problem (18), which allows us to separately optimize the
+downlink and uplink transmission resources.
+IV. OPTIMIZATION FRAMEWORK
+In this section, we specify the optimization framework
+for solving the uplink and downlink optimization problems,
+respectively.
+A. Uplink Optimization
+For the uplink aggregation, the optimization problem is
+minimize
+ζul,{rm},Θul
+ζul
+(19a)
+s.t.
+�
+l̸=m
+�
+j∈Kl
+ηl|rH
+m¯hm,j|2
+ηm|rH
+l ¯hl,j|2
+(19b)
++ ∥rm∥2(σul)2
+ηm
+≤ κmζul, ∀m ∈ M
+(19c)
+ζul ≥ 0.
+(19d)
+By setting optimzing varibales qi = ri/√ηi, ∀i ∈ M, the
+problem can be converted to
+minimize
+ζul,{qm},Θul
+ζul
+(20a)
+s.t.
+�
+l̸=m
+�
+j∈Kl
+|qH
+m¯hm,j|2
+|qH
+l ¯hl,j|2
++ (σul)2∥qH
+m∥2 ≤ κmζul, ∀m ∈ M
+(20b)
+|qH
+m¯hm,k|2 ≥ 1
+Pul
+, ∀m, ∀k ∈ Km
+(20c)
+(19d).
+
+Then we let
+|qH
+m¯hm,j|2
+|qH
+l ¯hl,j|2
+≤ bl,j, the optimization problem
+relaxes to
+minimize
+ζul,{qm,b},Θul
+ζul
+(21a)
+s.t.
+�
+l̸=m
+�
+j∈Kl
+bl,j + (σul)2∥qH
+m∥2 ≤ κmζul, ∀m
+(21b)
+|qH
+m¯hm,j|2
+|qH
+l ¯hl,j|2 ≤ bl,j, ∀l, j
+(21c)
+(19d), (20c).
+However, constraint (21c) is still non-convex, then we use
+the SCA method to transform (21c) into a linear con-
+straint which satisfies the property of convex. Let al,j =
+[ℜ(qH
+l ¯hl,j), ℑ(qH
+l ¯hl,j)], the corresponding approximated lin-
+ear constraint is
+|qH
+m¯hm,j|2
+bl,j
+≤ ∥al,j∥2
+≤ ∥a(t)
+l,j ∥2 + 2(a(t)
+l,j )T(al,j − a(t)
+l,j )
+(22)
+and
+∥a(t)
+m,k∥2 + 2(a(t)
+m,k)T(am,k − a(t)
+m,k) ≥ 1
+Pul
+.
+(23)
+The origin problem (21) is then approximated as
+minimize
+ζul,{qm,b,a},Θul
+ζul
+s.t.
+al,j = [ℜ(qH
+l ¯hl,j), ℑ(qH
+l ¯hl,j)], ∀l, j
+(19d), (21b), (22), (23).
+(24)
+And we can observe that the above problem turns out to be
+highly intractable due to the non-convexity of multiplication
+between variables q and Θul. Hence, a classical alternative
+optimization algorithm can be used to solve it.
+B. Downlink Optimization
+For the downlink dissemination, the optimization problem
+can be written as
+minimize
+ζdl,{tm},Θdl
+ζdl
+(25a)
+s.t.
+�
+k∈Km
+�
+l̸=m |¯hH
+l,ktl|2 + (σdl)2
+���¯hH
+m,ktm
+���
+2
+≤ κmζdl, ∀m
+(25b)
+∥tm∥2 ≤ P dl,
+(25c)
+ζdl ≥ 0.
+(25d)
+By letting
+�
+l̸=m |¯hH
+l,ktl|2+(σdl)2
+|¯hH
+m,ktm|
+2
+≤ dk, the optimization prob-
+lem is relaxed to
+minimize
+ζdl,{tm},Θdl
+ζdl
+(26a)
+s.t.
+�
+k∈Km
+dk ≤ κmζdl, ∀m ∈ M
+(26b)
+�
+l̸=m |¯hH
+l,ktl|2 + (σdl)2
+���¯hH
+m,ktm
+���
+2
+≤ dk,
+(26c)
+(25c), (25d).
+Similar as the uplink optimization, we can still convert
+(26c) to linear constraints using the SCA method. By setting
+cm,k = [ℜ(¯hH
+m,ktm), ℑ(¯hH
+m,ktm)], the relaxed problem is
+given as
+minimize
+ζdl,{tm,c},Θdl
+ζdl
+(27a)
+s.t.
+�
+l̸=m |¯hH
+l,ktl|2 + (σdl)2
+dk
+≤
+∥c(t)
+m,k∥2 + 2(c(t)
+m,k)T(cm,k − c(t)
+m,k), ∀m, k
+(27b)
+cm,k = [ℜ(¯hH
+m,ktm), ℑ(¯hH
+m,ktm)], ∀m, k
+(27c)
+(25c), (25d), (26b).
+The above problem (27) can be solved in the same way as
+uplink optimization.
+V. SIMULATION RESULTS
+In this section, we conduct extensive numerical experi-
+ments to evaluate the performance of the proposed SCA
+algorithm for the STAR-RIS assisted AirComp-based vertical
+FL system in multi-cell wireless network.
+We consider a STAR-RIS assisted two-cell wireless vertical
+FL network in a two-dimensional space, where the coordi-
+nates of the BSs are (0m, 0m) and (40m, 0m), the STAR-
+RIS is deployed at the edge of two cells, i.e., (20m, 0m).
+And the devices in each cell are uniformly located within a
+circular region centered at their corresponding BS with radius
+20 meters. All channel coefficients are modeled as
+h = ρ−α/2
+��
+β
+1 + β hLoS +
+�
+1
+1 + β hNLoS
+�
+(28)
+and vary independently over different rounds, where ρ denotes
+the distance between the transmitter and the receiver, α = 2.5
+denotes the pathloss exponent, β = 5 dB represents the Ri-
+cian factor, hLoS denotes the line-of-sight (LoS) component,
+and hNLoS denotes the non-line-of-sight (NLoS) exponent.
+In addition, the noise power are set to
+�
+σul�2 =
+�
+σdl�2 =
+−10dBm. All simulation results in the following are obtained
+by averaging over 100 experiments.
+We first evaluate the performance of uplink aggregation
+using AirComp and downlink dissemination error by consid-
+ering the MSE as the metric. As shown in Fig. 1, the MSE
+
+5
+10
+15
+20
+25
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+(a) MSE of AirComp versus the num-
+ber of elements at STAR-RIS when
+N = 8 and Km = 4.
+5
+10
+15
+20
+25
+0.01
+0.015
+0.02
+0.025
+0.03
+0.035
+0.04
+0.045
+(b) Downlink MSE versus the num-
+ber of elements at STAR-RIS when
+N = 8 and Km = 4.
+Fig. 1. Performance of uplink aggregation via AirComp under
+different settings.
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+(a) Training loss vs. Round
+10
+20
+30
+40
+50
+60
+70
+80
+90
+100
+40
+50
+60
+70
+80
+90
+100
+Average Testing Accuracy (%)
+(b) Testing accuracy vs. Round
+Fig. 2. Performance of AirComp assisted Vertical FL.
+decreases as the number of STAR-RIS elements increases for
+both downlink and uplink transmission, indicating that STAR-
+RIS can effectively enhance the signal transmission quality,
+particularly when it has a large number of elements.
+We further evaluate the performance of our proposed
+STAR-RIS assisted vertical FL system, where Km = 4
+devices in each cell cooperatively train a regularized logistic
+regression model. The number of antennas at each BS is
+N = 8, and the number of elements at STAR-RIS is Q = 10.
+We simulate the image classification task on Fashion-MNIST
+dataset [18]. And we assume that each cell perform a different
+binary classification task for simplify (0-1 in cell 1, 2-3 in
+cell 2). The traditional binary cross-entropy loss function is
+given as
+F(w) = − 1
+L
+L
+�
+i=1
+�
+yi �
+wxi�
+− ln
+�
+1 + exp(wxi)
+��
+.
+The learning rate µ(t) is set to 0.01.
+We consider the noiseless case as the performance upper
+bound. Fig. 2 shows that our proposed STAR-RIS assisted
+system converges quickly and achieves 96% testing accuracy
+in inference, which is far ahead compared with the other two
+cases. And it is even close to the performance upper bound.
+VI. CONCLUSION
+In this paper, we proposed a STAR-RIS assisted AirComp-
+based vertical FL system in multi-cell networks. To be
+specific, a STAR-RIS is deployed at the cell edge to facilitate
+the completion of different FL tasks by each cell. The Pareto
+boundary of the gap region is introduced to characterize
+the trade-off of learning performance among cells. We then
+formulate an optimization problem to minimize the sum of
+error-induced gaps across all cells, which is then solved by
+SCA-based algorithms. Our simulation results demonstrate
+that the proposed STAR-RIS assisted system can significantly
+improve the learning performance in both training and infer-
+ence phases thanks to its powerful capability of reducing the
+transmission errors.
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+arXiv:1708.07747, 2017.
+

ENE5T4oBgHgl3EQfUg_2/content/tmp_files/load_file.txt

@@ -0,0 +1,470 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf,len=469
+page_content='STAR-RIS Assisted Over-the-Air Vertical Federated  Learning in Multi-Cell Wireless Networks  Xiangyu Zeng∗†‡, Yijie Mao∗, and Yuanming Shi∗  ∗School of Information Science and Technology, ShanghaiTech University, Shanghai 201210, China  †Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, China  ‡University of Chinese Academy of Sciences, Beijing 100049, China  E-mail: {zengxy, maoyj, shiym}@shanghaitech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='cn  Abstract—Vertical federated learning (FL) is a critical enabler  for distributed artificial intelligence services in the emerging  6G era, as it allows for secure and efficient collaboration of  machine learning among a wide range of Internet of Things  devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' However, current studies of wireless FL typically con-  sider a single task in a single-cell wireless network, ignoring  the impact of inter-cell interference on learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  In this paper, we investigate a simultaneous transmitting and  reflecting reconfigurable intelligent surface (STAR-RIS) assisted  over-the-air computation based vertical FL system in multi-cell  networks, in which a STAR-RIS is deployed at the cell edge  to facilitate the completion of different FL tasks in different  cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' We establish the convergence of the proposed system  through theoretical analysis and introduce the Pareto boundary  of the optimality gaps to characterize the trade-off among cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Based on the analysis, we then jointly design the transmit and  receive beamforming as well as the STAR-RIS transmission and  reflection coefficient matrices to minimize the sum of the gaps of  all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' To solve the non-convex resource allocation problem, we  introduce a successive convex approximation based algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Numerical experiments demonstrate that compared with con-  ventional approaches, the proposed STAR-RIS assisted vertical  FL model and the cooperative resource allocation algorithm  achieve much lower mean-squared error for both uplink and  downlink transmission in multi-cell wireless networks, resulting  in improved learning performance for vertical FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' INTRODUCTION  Federated learning (FL) is a machine learning (ML) ap-  proach that enables multiple parties to collaboratively train a  learning model without revealing their individual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' This  is beneficial in a variety of fields where data privacy is a  concern, as FL allows parties to maintain control over their  own data while still benefiting from the combined knowledge  of all parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' In modern wireless Internet of Things (IoT)  networks, data is often collected from various types of devices  [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' To facilitate data analysis in such settings, vertical FL,  a variation of FL that is designed to address the challenges  of training machine learning models on vertically partitioned  data silos, is commonly adopted [2]–[6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  One major issue that prevents the implementation of (ver-  tical) FL in real-world application is the communication  latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' To address this issue, over-the-air computation (Air-  Comp) has been proposed to facilitate fast wireless data  aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' By utilizing the superposition property of wire-  less multiple access channels (MAC) to concurrently transmit  and aggregate local updates, AirComp significantly reduces  communication latency compared to orthogonal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Previous research has explored the use of AirComp in FL,  such as the joint design of device selection and beamforming  for fast global model aggregation in [5], and the development  of a broadband analog aggregation scheme for low latency FL  with linear growth of latency reduction ratio in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  On the other hand, the coexistence of multiple FL tasks  in multi-cell networks has yet to be fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Though  the authors in [8] have studied the bandwidth allocation  for multiple FL tasks, the system model is limited to a  single-cell network and the impact of inter-cell interference  on FL performance remains unplumbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' It has been well  investigated that reconfigurable intelligent surface (RIS), a  metasurface composed of reconfigurable passive elements,  can modify the propagation environment of wireless signal  and reduce multi-cell interference [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' However, conventional  RISs are reflecting only with limited wireless coverage [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  The recently introduced simultaneous transmitting and reflect-  ing RIS (STAR-RIS), which allows the source and destination  to be located at either side of the metasurface, has been  recognized as a promising strategy to enhance the coverage  of each cell and further reduce inter-cell interference [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  STAR-RIS is therefore a promising technique to facilitate FL  in multi-cell networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' To the best of our knowledge, STAR-  RIS assisted vertical FL has not been studied yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  In this paper, inspired by the benefits of AirComp for global  aggregation [12] and the merits of STAR-RIS in multi-cell  networks, we fill the research gap and propose a STAR-RIS  assisted AirComp-based vertical FL in multi-cell networks,  where a STAR-RIS is deployed at the cell edge to assist each  cell in completing different FL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Through theoretical  analysis, we demonstrate the convergence of our proposed  vertical FL process and introduce the Pareto boundary of the  gap region to characterize the trade-off performance among  multiple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' This allows us to formulate an optimization  problem with the aim of minimizing the sum of error-induced  gaps for all cells using the proposed algorithm based on suc-  cessive convex approximation (SCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Numerical experiments  confirm the validity of our theoretical analysis and show the  superiority of our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' SYSTEM MODEL  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Learning Framework  Consider a STAR-RIS assisted multi-cell wireless net-  work consisting of M base stations (BS) with N an-  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='05545v1  [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='IT]  13 Jan 2023  tennas, where BS m  ∈  M  =  {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , M} aims  to  train  an  ML  model  by  coordinating  Km  single-  antenna devices located in cell m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Specifically, device  k ∈ Km =  ��m−1  l=1 Kl + 1, �m−1  l=1 Kl + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , �m−1  l=1 Kl+  Km} is associated with BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And there is one STAR-RIS  equipped with Q passive reflecting/transmitting elements, de-  ployed at the cell-edge of all cells to boost the signal strength  of edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Each cell is equipped with a vertically  partitioned dataset, where different devices hold different  features of the same samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' For simplicity, we assume that  each cell has the same number of samples and that devices  within each cell contain the same number of non-overlapping  features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Let Dm = {(xi  m,1, · · · , xi  m,Km), yi  m}Lm  i=1 denote the  whole training dataset of Lm samples in cell m, where xi  m,k  denotes the partial features of sample i located at device k in  cell m, and yi  m denotes the corresponding label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' In vertical  FL, it is assumed that the BS holds all labels ym = {yi  m}Lm  i=1,  and device k is only available to its own local feature set  Dm,k = {xi  m,k}Lm  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And xi  m = [(xi  m,1)T, · · · , (xi  m,Km)T]T  denotes the overall feature vector of sample i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  The goal of vertical FL in cell m is to collaboratively learn  a global model wm (concatenated vector of wk for k ∈ Km)  that maps an input to the corresponding prediction through  a continuously differentiable function σ(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Since features of  one sample are distributed at different devices, we assume  that device k maps the local feature xk to local prediction  result gk(wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' This paper considers a linear form for the  local prediction function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=', gk(wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' xk) = wT  k xk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' By ag-  gregating local prediction results, the final prediction in cell m  can be obtained by σ(wm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' xm) = σ(�  k∈Km gk(wk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' xk)) =  σ(wT  mxm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' In order to learn the global model wm in cell m,  we propose to minimize the loss function as  min  wm F(wm) =  1  Lm  Lm  �  i=1  f  �  σ(wT  mxi  m);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi  m  �  ,  (1)  where f(·) is the sample-wise loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  In our multi-cell system, each cell performs a unique FL  task using the full batch gradient descent (GD) approach,  which is described in the following subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' We assume  universal frequency reuse, meaning that all cells share the  same frequency channel, leading to inter-cell interference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' GD Algorithm for Vertical FL  In this subsection, we introduce the framework of GD  algorithm for vertical FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' For brevity, the subscript of cell m  is omitted for Lm, wm, xm, ym.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The GD algorithm specified  in this subsection is applied for all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Let ∇F(w) denote  the gradient of F respect to w, which is calculated as  ∇F(w) = 1  L  L  �  i=1  ∇f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi),  (2)  where  ∇f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi)  denote  the  gradient  of  f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi) respect to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Based on the chain rule,  the gradient of f is rewritten as  ∇f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi) = G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi)xi,  (3)  where G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi) = ∂f(σ(wTxi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi)/∂wTxi is an auxil-  iary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Hence, ∇F(w) can be rewritten as  ∇F(w) = 1  L  L  �  i=1  G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi)xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (4)  Recall that the BS holds all labels y, so G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi) can  be calculated at the BS only if the BS can access the  aggregation of local predictions {wTxi}L  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Specifically, at  the t-th communication round, the BS and the edge devices  in each cell perform the following three procedures:  Broadcasting: The BS computes {G((w(t))Txi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi)}L  i=1  and broadcasts the result back to its corresponding devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Local model update: After broadcasting, device k com-  putes the partial gradient ∇kF(wk) with local data Dk, given  as  ∇kF(wk) = 1  L  L  �  i=1  G(wTxi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' yi)xi  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (5)  Each device can thus update its local model by taking a step  of GD with learning rate µ(t) as  w(t+1)  k  = w(t)  k  − µ(t)∇kF(w(t)  k ),  (6)  where w(t)  k  is the local model of device k at the t-th round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Local prediction and global aggregation: device k com-  putes the local prediction results {(w(t+1)  k  )Txi  k}L  i=1 and  sends to the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And BS aggregates them to get final  prediction result {(w(t+1))Txi}L  i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Since the BS only needs the aggregation of local prediction  results, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=', neither local features nor local models need be  uploaded to the BS, which significantly enhances privacy pro-  tection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' In addition, the communication efficiency is improved  since the local prediction result is usually low-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Communication Model  In this subsection, the proposed communication model is  delineated with a special focus on the STAR-RIS assisted  uplink and downlink transmission models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  1) STAR-RIS: The STAR-RIS is a type of RIS that can  produce omnidirectional radiation by implementing equiva-  lent electric and magnetic currents in its hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' It has  three protocols for use in wireless networks: energy splitting,  mode switching, and time switching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' In this article, we focus  on the mode-switching protocol, in which each element of  the STAR-RIS can operate in either the reflection mode (R  mode) or the transmission mode (T mode).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Such on-off type  of operating protocol is simpler to implement compared to  the energy splitting protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Specifically, one group consists  of Qt elements operating in the T mode, while the other  group contains Qr elements operating in the R mode, where  Qt + Qr = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Accordingly, the STAR-RIS transmission-  coefficient  and  reflection-coefficient  matrices  are  given  by Θt  = diag  ��  βt  1ejθt  1,  �  βt  2ejθt  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' ,  �  βt  Qejθt  Q  �  and  Θr  = diag  ��  βr  1ejθr  1,  �  βr  2ejθr  2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' ,  �  βr  Qejθr  Q  �  , respec-  tively, where βt  q, βr  q ∈ {0, 1}, βt  q + βr  q = 1, and θt  q, θr  q ∈  [0, 2π), ∀q ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , Q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The M cells can be divided  into two groups Mr and Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Specifically, cell m is in the  reflection dimension with m ∈ Mr and in the transmission  dimension with m ∈ Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Let hm,k ∈ CN, hr  k ∈ CQ and Gm ∈ CQ×N denote the  equivalent channels from edge device k to BS m, from edge  device k to the STAR-RIS, and from the STAR-RIS to BS  m, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The combined channel from the k-th edge  device to the BS m via the STAR-RIS can be written as  ¯hm,k =  � hm,k + GH  mΘthr  k, ∀m ∈ Mt,  hm,k + GH  mΘrhr  k, ∀m ∈ Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Note that the uplink and downlink STAR-RIS matrices can be  separatively designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' For simplify, we write Θt and Θr for  uplink and downlink transmission in terms of Θul and Θdl,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  2) Uplink transmission: In the uplink transmission, we  assume the devices communicate with the BS via AirComp,  which has a wide range of FL applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Specifically, we denote sk = [s1  k, s2  k, · · · , sL  k ]T ∈ CL  as the local prediction results at device k, where the local  prediction result of the i-th sample si  k = wT  k xi  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' At each  time slot i ∈ {1, 2, · · · , Lm}, each device in cell m sends  the corresponding prediction result of the i-th sample to BS  m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And we assume that sk is normalized with zero mean  and unit variance [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' We denote gm(i) = �  k∈Km si  k as the  target function to be estimated through AirComp at the i-th  time slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' To simplify the notation, we omit the time index by  writing g(i) and si  k as g and sul  k , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And we assume  that the signals transmitted by all devices are synchronized  at the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Then the received signal at BS m is given by  yul  m =  �  k  ¯hm,kbksul  k + nul  m,  (7)  where bk ∈ C is the transmit scalar at device k, and nul  m is the  additive white Gaussian noise with zero mean and variance  (σul)2 at BS m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The transmit power constraint at device k is  E(|bksul  k |2) = |bk|2 ≤ P ul, where P ul > 0 is the maximum  transmit power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The scaled signal received at BS m is  ¯gm =  1  √ηm  rH  myul  m =  1  √ηm  rH  m  �  k∈K  ¯hm,kbksul  k + rH  mnul  m  √ηm  , (8)  where rm ∈ CN is the receive beamforming vector and ηm  is a normalizing factor for cell m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' To compensate for the  phase distortion introduced by complex channel responses,  the transmit scalar at device k in cell m is set to bk =  √ηm  (rH  m¯hm,k)H  |rHm¯hm,k|2 , ∀k ∈ Km, and ηm can be expressed as  ηm = P ul mink∈Km |rH  m¯hm,k|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Then the estimated function  at BS for cell m is given as  ˆgm = ℜ{¯gm}  = ℜ{gm +  1  √ηm  rH  m  �  l̸=m  �  j∈Kl  ¯hm,jbjsul  j + rH  mnul  m  √ηm  �  ��  �  eulm  }  = gm + ℜ{eul  m}  (9)  3) Downlink transmission: After obtaining the estimate  ˆgm in the cell m, BS m computes G(ˆgm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' y) with noisy  aggregation ˆgm, and then broadcasts the result to the associ-  ated devices in Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And we write G(ˆgm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' y) in terms of Gm  for simplify.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Without loss of generality, we assume that the  transmitted signal follows the standard Gaussian distribution,  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=', Gm ∼ CN(0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The received signal at device k is  ydl  k =  �  m  ¯hH  m,ktmGm + ndl  k ,  (10)  where tm denotes the transmit beamforming vector at BS  m, and ndl  k ∼ CN  �  0, (σdl)2�  is the additive white Gaus-  sian noise with zero mean and variance (σdl)2 at device  k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The maximum transmit power at BS m is P dl, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=',  E(∥Gmtm∥2) = ∥tm∥2 ≤ P dl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  To compensate for the phase distortion introduced by  complex channel responses, the receive scalar at device k  in cell m is set to rk =  (¯hH  m,ktm)H  |¯hH  m,ktm|  2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The estimated Gm at  device k is given as  ˆGm,k = ℜ{rkydl  k }  = ℜ{Gm +  (¯hH  m,ktm)H  ���¯hH  m,ktm  ���  2  �  ��  l̸=m  ¯hH  l,ktlGl + ndl  k  �  �  �  ��  �  ¯edl  k  }  = Gm + ℜ{¯edl  k }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (11)  Note that the uplink noise is embedded in function Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' In  order to directly describe the effective noise, we expand Gm  to its first-order Taylor expansion as follows  ˆGm,k  = Gm + ℜ{¯edl  k }  = G(gm + ℜ{eul  m};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' y) + ℜ{¯edl  k }  = G(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' y) + G  ′(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' y)ℜ{eul  m} + O(|ℜ{eul  m}|2) + ℜ{¯edl  k }  ≈ G(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' y) + G  ′(gm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' y)ℜ{eul  m} + ℜ{¯edl  k }  �  ��  �  edl  k  ,  (12)  where G′(·) is the first derivative of G(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Assume that the  noise amplitude is small, the term O(|ℜ{eul  m}|2) is neglected,  which implies the last approximation in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' CONVERGENCE ANALYSIS AND PROBLEM  FORMULATION  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Convergence Analysis  In previous work [12], [14], [15], the convergence analysis  of the AirComp-based vertical FL process in each cell has  been established under the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Assumption 1 (α-strongly convexity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The function F(·) is  assumed to be α-strongly convex on Rd with constant α,  namely, for all x, y ∈ Rd, we have  F(y) ≥ F(x) + ∇F(x)T(y − x) + α  2 ∥y − x∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Assumption 2 (β-smoothness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The function F(·) is assumed  to be β-smooth on Rd with constant β, namely, for all x, y ∈  Rd, we have  F(y) ≤ F(x) + ∇F(x)T(y − x) + β  2 ∥y − x∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Theorem 1 (Convergence of vertical FL process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Suppose  that Assumption 1 and 2 hold, setting the learning rate to  be 0 < µ(t) ≤  1  β , then the expected optimality gap after T  communication rounds is upper bounded by  E  �  F(w(T )  m ) − F(w∗  m)  �  ≤ ρT E  �  F(w(0)  m ) − F(w∗  m)  �  +  1  2βL2  T −1  �  t=0  ρT −t−1 �  k∈Km  �  Φ1,kE[|ℜ{eul  m}|2] + Φ2,kE[|ℜ{¯edl  k }|2]  �  ,  (13)  where ρ = 1−α/β, Φ1,k = �L  i=1 ∥(Gi  m,k)  ′xi  k∥2  2 and Φ2,k =  �L  i=1 ∥xi  k∥2  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Please refer to previous work [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Problem Formulation  According to Theorem 1, the convergence optimality gap  is largely determined by the mean-squared-error (MSE) of  both gm and Gm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' However, solely optimizing MSE for each  cell through AirComp may result in significant inter-cell  interference in the considered multi-cell wireless networks,  which can negatively impact the learning performance of  other cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' As such, it is necessary to carefully balance the  learning performance among various FL tasks in multiple  cells through a cooperative design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  We begin by identifying the gap region G, to be the set of  tuples (∆1, ∆2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , ∆M), which represents the instantaneous  errors that cause gaps in all cells, and can be achieved simul-  taneously under specific downlink and uplink transmission  power constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The gap region G can be represented as  G =  �  {(∆1, ∆2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , ∆M)|∆m ≥ Gapm, ∀m ∈ M}, (14)  where  Gapm =  �  k∈Km  �  Φ1,kE[|ℜ{eul  m}|2] + Φ2,kE[|ℜ{¯edl  k }|2]  �  ,  (15)  E[|ℜ{eul  m}|2] =  �  l̸=m,j∈Kl  ηl|rH  m¯hm,j|2  ��m|rH  l ¯hl,j|2 + ∥rm∥2σ2  ul  ηm  ,  E[|ℜ{¯edl  k }|2] =  �  l̸=m |¯hH  l,ktl|2 + (σdl)2  ���¯hH  m,ktm  ���  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (16)  As previously stated, in order to decrease the error-induced  gap in one cell, the gaps of other cells maybe increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  In light of this, our objective is to find a suitable solution  that allows us to achieve the Pareto boundary of the gap  region G, so as to balance the performance of learning among  multiple cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' In this context, the Pareto optimality of a tuple  is described as follows [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Here, we leverage the profiling technique [17] to char-  acterize the Pareto boundary by coordinating all BSs to  minimize the sum of Gap of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Specifically, let  κ = [κ1, κ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , κM] denote a given profiling vector, which  satisfies κm ≥ 0, ∀m ∈ M, and �  m∈M κm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The gap  tuple on Pareto boundary can be obtained by solving the  following problem  minimize  ζ,{rm},{tm},Θt,Θr  ζ  (17a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Gapm ≤ κmζ, ∀m ∈ M  (17b)  ζ ≥ 0,  (17c)  where ζ denotes the sum of the gaps of all cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Thus,  the gap tuple can be represented as (∆1, ∆2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , ∆M) =  (κ1ζ, κ2ζ, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' , κMζ), where a smaller value of κm implies a  more stringent requirement for the gap of cell m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Denote ζ = ζul +ζdl, where ζul and ζdl are used to quan-  tify the sum of instantaneous error-induced gaps generated by  uplink and downlink transmissions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Hence, we  rewrite problem (17) as  minimize  ζul,ζdl,{rm},{tm},Θt,Θr  ζul + ζdl  (18a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Gapul  m ≤ κmζul, ∀m ∈ M  (18b)  Gapdl  m ≤ κmζdl, ∀m ∈ M  (18c)  ζul ≥ 0  (18d)  ζdl ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (18e)  The downlink and uplink transmissions can be decoupled in  problem (18), which allows us to separately optimize the  downlink and uplink transmission resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' OPTIMIZATION FRAMEWORK  In this section, we specify the optimization framework  for solving the uplink and downlink optimization problems,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Uplink Optimization  For the uplink aggregation, the optimization problem is  minimize  ζul,{rm},Θul  ζul  (19a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  �  l̸=m  �  j∈Kl  ηl|rH  m¯hm,j|2  ηm|rH  l ¯hl,j|2  (19b)  + ∥rm∥2(σul)2  ηm  ≤ κmζul, ∀m ∈ M  (19c)  ζul ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (19d)  By setting optimzing varibales qi = ri/√ηi, ∀i ∈ M, the  problem can be converted to  minimize  ζul,{qm},Θul  ζul  (20a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  �  l̸=m  �  j∈Kl  |qH  m¯hm,j|2  |qH  l ¯hl,j|2  + (σul)2∥qH  m∥2 ≤ κmζul, ∀m ∈ M  (20b)  |qH  m¯hm,k|2 ≥ 1  Pul  , ∀m, ∀k ∈ Km  (20c)  (19d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Then we let  |qH  m¯hm,j|2  |qH  l ¯hl,j|2  ≤ bl,j, the optimization problem  relaxes to  minimize  ζul,{qm,b},Θul  ζul  (21a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  �  l̸=m  �  j∈Kl  bl,j + (σul)2∥qH  m∥2 ≤ κmζul, ∀m  (21b)  |qH  m¯hm,j|2  |qH  l ¯hl,j|2 ≤ bl,j, ∀l, j  (21c)  (19d), (20c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  However, constraint (21c) is still non-convex, then we use  the SCA method to transform (21c) into a linear con-  straint which satisfies the property of convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Let al,j =  [ℜ(qH  l ¯hl,j), ℑ(qH  l ¯hl,j)], the corresponding approximated lin-  ear constraint is  |qH  m¯hm,j|2  bl,j  ≤ ∥al,j∥2  ≤ ∥a(t)  l,j ∥2 + 2(a(t)  l,j )T(al,j − a(t)  l,j )  (22)  and  ∥a(t)  m,k∥2 + 2(a(t)  m,k)T(am,k − a(t)  m,k) ≥ 1  Pul  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (23)  The origin problem (21) is then approximated as  minimize  ζul,{qm,b,a},Θul  ζul  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  al,j = [ℜ(qH  l ¯hl,j), ℑ(qH  l ¯hl,j)], ∀l, j  (19d), (21b), (22), (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (24)  And we can observe that the above problem turns out to be  highly intractable due to the non-convexity of multiplication  between variables q and Θul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Hence, a classical alternative  optimization algorithm can be used to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Downlink Optimization  For the downlink dissemination, the optimization problem  can be written as  minimize  ζdl,{tm},Θdl  ζdl  (25a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  �  k∈Km  �  l̸=m |¯hH  l,ktl|2 + (σdl)2  ���¯hH  m,ktm  ���  2  ≤ κmζdl, ∀m  (25b)  ∥tm∥2 ≤ P dl,  (25c)  ζdl ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  (25d)  By letting  �  l̸=m |¯hH  l,ktl|2+(σdl)2  |¯hH  m,ktm|  2  ≤ dk, the optimization prob-  lem is relaxed to  minimize  ζdl,{tm},Θdl  ζdl  (26a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  �  k∈Km  dk ≤ κmζdl, ∀m ∈ M  (26b)  �  l̸=m |¯hH  l,ktl|2 + (σdl)2  ���¯hH  m,ktm  ���  2  ≤ dk,  (26c)  (25c), (25d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Similar as the uplink optimization, we can still convert  (26c) to linear constraints using the SCA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' By setting  cm,k = [ℜ(¯hH  m,ktm), ℑ(¯hH  m,ktm)], the relaxed problem is  given as  minimize  ζdl,{tm,c},Θdl  ζdl  (27a)  s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  �  l̸=m |¯hH  l,ktl|2 + (σdl)2  dk  ≤  ∥c(t)  m,k∥2 + 2(c(t)  m,k)T(cm,k − c(t)  m,k), ∀m, k  (27b)  cm,k = [ℜ(¯hH  m,ktm), ℑ(¯hH  m,ktm)], ∀m, k  (27c)  (25c), (25d), (26b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  The above problem (27) can be solved in the same way as  uplink optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' SIMULATION RESULTS  In this section, we conduct extensive numerical experi-  ments to evaluate the performance of the proposed SCA  algorithm for the STAR-RIS assisted AirComp-based vertical  FL system in multi-cell wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  We consider a STAR-RIS assisted two-cell wireless vertical  FL network in a two-dimensional space, where the coordi-  nates of the BSs are (0m, 0m) and (40m, 0m), the STAR-  RIS is deployed at the edge of two cells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=', (20m, 0m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  And the devices in each cell are uniformly located within a  circular region centered at their corresponding BS with radius  20 meters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' All channel coefficients are modeled as  h = ρ−α/2  ��  β  1 + β hLoS +  �  1  1 + β hNLoS  �  (28)  and vary independently over different rounds, where ρ denotes  the distance between the transmitter and the receiver, α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='5  denotes the pathloss exponent, β = 5 dB represents the Ri-  cian factor, hLoS denotes the line-of-sight (LoS) component,  and hNLoS denotes the non-line-of-sight (NLoS) exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  In addition, the noise power are set to  �  σul�2 =  �  σdl�2 =  −10dBm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' All simulation results in the following are obtained  by averaging over 100 experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  We first evaluate the performance of uplink aggregation  using AirComp and downlink dissemination error by consid-  ering the MSE as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' 1, the MSE  5  10  15  20  25  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='06  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='08  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='12  (a) MSE of AirComp versus the num-  ber of elements at STAR-RIS when  N = 8 and Km = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  5  10  15  20  25  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='01  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='015  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='02  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='025  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='03  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='035  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='04  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='045  (b) Downlink MSE versus the num-  ber of elements at STAR-RIS when  N = 8 and Km = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Performance of uplink aggregation via AirComp under  different settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  10  20  30  40  50  60  70  80  90  100  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='7  (a) Training loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Round  10  20  30  40  50  60  70  80  90  100  40  50  60  70  80  90  100  Average Testing Accuracy (%)  (b) Testing accuracy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Round  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Performance of AirComp assisted Vertical FL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  decreases as the number of STAR-RIS elements increases for  both downlink and uplink transmission, indicating that STAR-  RIS can effectively enhance the signal transmission quality,  particularly when it has a large number of elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  We further evaluate the performance of our proposed  STAR-RIS assisted vertical FL system, where Km = 4  devices in each cell cooperatively train a regularized logistic  regression model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The number of antennas at each BS is  N = 8, and the number of elements at STAR-RIS is Q = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  We simulate the image classification task on Fashion-MNIST  dataset [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And we assume that each cell perform a different  binary classification task for simplify (0-1 in cell 1, 2-3 in  cell 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The traditional binary cross-entropy loss function is  given as  F(w) = − 1  L  L  �  i=1  �  yi �  wxi�  − ln  �  1 + exp(wxi)  ��  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  The learning rate µ(t) is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  We consider the noiseless case as the performance upper  bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' 2 shows that our proposed STAR-RIS assisted  system converges quickly and achieves 96% testing accuracy  in inference, which is far ahead compared with the other two  cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' And it is even close to the performance upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' CONCLUSION  In this paper, we proposed a STAR-RIS assisted AirComp-  based vertical FL system in multi-cell networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' To be  specific, a STAR-RIS is deployed at the cell edge to facilitate  the completion of different FL tasks by each cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' The Pareto  boundary of the gap region is introduced to characterize  the trade-off of learning performance among cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' We then  formulate an optimization problem to minimize the sum of  error-induced gaps across all cells, which is then solved by  SCA-based algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Our simulation results demonstrate  that the proposed STAR-RIS assisted system can significantly  improve the learning performance in both training and infer-  ence phases thanks to its powerful capability of reducing the  transmission errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content='  REFERENCES  [1] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
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+page_content='  [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
+page_content=' Letaief, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE5T4oBgHgl3EQfUg_2/content/2301.05545v1.pdf'}
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+arXiv:2301.02890v1  [math.DS]  7 Jan 2023
+ERGODICITY AND PERIODIC ORBITS OF p-ADIC (1, 2)-RATIONAL
+DYNAMICAL SYSTEMS WITH TWO FIXED POINTS
+I.A. SATTAROV, E.T. ALIEV
+Abstract. We consider (1, 2)-rational functions given on the field of p-adic numbers Qp.
+In general, such a function has four parameters. We study the case when such a function
+has two fixed points and show that when there are two fixed points then (1, 2)-rational
+function is conjugate to a two-parametric (1, 2)-rational function.
+Depending on these
+two parameters we determine type of the fixed points, find Siegel disks and the basin of
+attraction of the fixed points. Moreover, we classify invariant sets and study ergodicity
+properties of the function on each invariant set. We describe 2- and 3-periodic orbits of
+the p-adic dynamical systems generated by the two-parametric (1, 2)-rational functions.
+1. Introduction and preliminaries
+A function is called a (n, m)-rational function if and only if it can be written in the
+form f(x) = Pn(x)
+Qm(x), where Pn(x) and Qm(x) are polynomial functions with degree n and m
+respectively, Qm(x) is non zero polynomial.
+In this paper we study dynamical systems generated by a (1.2-)rational function. Our
+investigations based on methods of [1], [3], [13]-[17]. For motivations of the study see [2],
+[4]-[6], [10]-[12] and the references therein.
+Let us give main definitions. Let Q be the field of rational numbers. The greatest common
+divisor of the positive integers n and m is denotes by (n, m). Every rational number x ̸= 0
+can be represented in the form x = pr n
+m, where r, n ∈ Z, m is a positive integer, (p, n) = 1,
+(p, m) = 1 and p is a fixed prime number.
+The p-adic norm of x ∈ Q is given by
+|x|p =
+�
+p−r,
+for x ̸= 0,
+0,
+for x = 0.
+It has the following properties:
+1) |x|p ≥ 0 and |x|p = 0 if and only if x = 0,
+2) |xy|p = |x|p|y|p,
+3) the strong triangle inequality
+|x + y|p ≤ max{|x|p, |y|p},
+3.1) if |x|p ̸= |y|p then |x + y|p = max{|x|p, |y|p},
+2010 Mathematics Subject Classification. 46S10, 12J12, 11S99, 30D05, 54H20.
+Key words and phrases. Rational dynamical systems; fixed point; invariant set; Siegel disk; complex
+p-adic field.
+1
+
+2
+I.A. SATTAROV, E.T. ALIEV
+3.2) if |x|p = |y|p then |x + y|p ≤ |x|p.
+The completion of Q with respect to p-adic norm defines the p-adic field which is denoted
+by Qp (see [8]).
+For any a ∈ Qp and r > 0 denote
+Ur(a) = {x ∈ Qp : |x − a|p < r},
+Vr(a) = {x ∈ Qp : |x − a|p ≤ r},
+Sr(a) = {x ∈ Qp : |x − a|p = r}.
+A function f : Ur(a) → Qp is said to be analytic if it can be represented by
+f(x) =
+∞
+�
+n=0
+fn(x − a)n,
+fn ∈ Qp,
+which converges uniformly on the ball Ur(a).
+Now let f : U → U be an analytic function. Denote f n(x) = f ◦ · · · ◦ f
+�
+��
+�
+n
+(x).
+If f(x0) = x0 then x0 is called a fixed point. The set of all fixed points of f is denoted
+by Fix(f). A fixed point x0 is called an attractor if there exists a neighborhood U(x0) of
+x0 such that for all points x ∈ U(x0) it holds lim
+n→∞ f n(x) = x0. If x0 is an attractor then its
+basin of attraction is
+A(x0) = {x ∈ Qp : f n(x) → x0, n → ∞}.
+A fixed point x0 is called repeller if there exists a neighborhood U(x0) of x0 such that
+|f(x) − x0|p > |x − x0|p for x ∈ U(x0), x ̸= x0.
+Let x0 be a fixed point of a function f(x). Put λ = f ′(x0). The point x0 is attractive if
+0 < |λ|p < 1, indifferent if |λ|p = 1, and repelling if |λ|p > 1.
+The ball Ur(x0) is said to be a Siegel disk if each sphere Sρ(x0), ρ < r is an invariant
+sphere of f(x), i.e. if x ∈ Sρ(x0) then all iterated points f n(x) ∈ Sρ(x0) for all n = 1, 2 . . . .
+The union of all Siegel desks with the center at x0 is said to a maximum Siegel disk and is
+denoted by SI(x0).
+Let f : A → A and g : B → B be two maps. f and g are said to be topologically conjugate
+if there exists a homeomorphism h : A → B such that, h◦f = g ◦h. The homeomorphism h
+is called a topological conjugacy. Mappings which are topologically conjugate are completely
+equivalent in terms of their dynamics.
+In this paper we consider (1, 2)-rational function f : Qp → Qp defined by
+f(x) =
+ax + b
+x2 + cx + d,
+x ̸= ˆx1,2 = −c ±
+√
+c2 − 4d
+2
+(1.1)
+where the parameters of the function satisfy the following conditions
+a ̸= 0,
+a, b, c, d,
+�
+c2 − 4d ∈ Qp.
+We study p-adic dynamical systems generated by the rational function (1.1). The equa-
+tion f(x) = x for fixed points of the function (1.1) is equivalent to the equation
+x3 + cx2 + (d − a)x − b = 0.
+(1.2)
+The equation (1.2) may have three solutions with one of the following relations:
+
+ERGODICITY AND PERIODIC ORBITS
+3
+(i) one solution having multiplicity three;
+(ii) two solutions, one of which has multiplicity two;
+(iii) three distinct solutions.
+Remark 1. Since the behavior of dynamical system depends on the set of fixed points, each
+of the above mentioned case (i)-(iii) has its own character of dynamics. In [15] the case (i)
+was considered. In this paper we consider the case (ii), i.e., we investigate the behavior of
+the trajectories of an arbitrary (1, 2)-rational dynamical system in Qp when there are two
+fixed points for f. The case (iii) will be considered in a separate paper.
+The paper is organized as follows. In Section 2 under some assumptions we show that
+four-parametric function (1.1) is conjugate to a two-parametric (1,2)-rational function. In
+Section 3 we study the p-adic dynamics generated by the two-parametric function and give
+Siegel disks, the basin of attractions and classification of all invariant sets. In Section 4 we
+investigate ergodicity of this dynamical systems on invariant sets. In Section 5 we describe
+2- and 3-periodic orbits.
+2. A function conjugate to (1.1)
+Denote by x1 and x2 the two solutions of the equation (1.2), where x2 has multiplicity
+two. Then we have x3 + cx2 + (d − a)x − b = (x − x1)(x − x2)2 and
+
+
+
+
+
+x1 + 2x2 = −c
+x2
+2 + 2x1x2 = d − a
+x1x2
+2 = b.
+(2.1)
+Let homeomorphism h : Qp → Qp be defined by h(t) = t+x2. We note that, the function
+f is topologically conjugate to function h−1 ◦ f ◦ h. We have
+(h−1 ◦ f ◦ h)(t) = −x2t2 + Bt
+t2 + Dt + B ,
+(2.2)
+where B = x2
+2 + cx2 + d and D = 2x2 + c.
+In [13] the case x2 ̸= 0 is studied.
+Thus in this paper we consider the case x2 = 0 in (2.2). If x2 = 0, then B = d = a and
+D = c. Thus we have the following proposition
+Proposition 1. Any (1,2)-rational function having two distinct fixed points is topologically
+conjugate to one of the following functions
+f(x) =
+ax2 + bx
+x2 + cx + b,
+ab(a − c) ̸= 0,
+a, b, c ∈ Qp,
+and
+f(x) =
+ax
+x2 + cx + a,
+ac ̸= 0,
+a, c, ∈ Qp.
+(2.3)
+where x ̸= ˆx1,2 = −c±
+√
+c2−4a
+2
+.
+We study the dynamical system (Qp, f) with f given by (2.3).
+
+4
+I.A. SATTAROV, E.T. ALIEV
+3. p-Adic dynamics of (2.3)
+Note that, the function (2.3) has two fixed points x1 = 0 and x2 = −c. We have
+f ′(x1) = 1 and f ′(x2) = 1 − c2
+a .
+Thus, the point x1 is an indifferent point for (2.3), i.e., x1 is a center of some Siegel disk
+SI(x1). In this section we determine the character of the fixed point x2 for each cases.
+Then we find Siegel disk or basin of attraction of the fixed point x2, when x2 is indifferent
+or attractive, respectively. In the case where x2 is repelling, we find open ball Ur(x2), such
+that the inequality |f(x) − x2|p > |x − x2|p holds for all x ∈ Ur(x2). Moreover, we study a
+relation between the sets SI(x1) and SI(x2) when x2 is an indifferent.
+For any x ∈ Qp, x ̸= ˆx1,2, by simple calculations we get
+|f(x)|p = |x|p ·
+|a|p
+|x − ˆx1|p|x − ˆx2|p
+.
+(3.1)
+Denote
+P = {x ∈ Qp : ∃n ∈ N ∪ {0}, f n(x) ∈ {ˆx1, ˆx2}},
+α = min{|ˆx1|p, |ˆx2|p} and β = max{|ˆx1|p, |ˆx2|p}.
+(3.2)
+Since ˆx1 + ˆx2 = −c, we have |c|p ≤ α for α = β and |c|p = β for α < β. Also, since
+ˆx1ˆx2 = a, we have |a|p = αβ.
+Theorem 1. The p-adic dynamical system generated by the function (2.3) has the following
+properties:
+1. SI(x1) = Uα(0).
+2. If |c|p < α = β, then x2 is indifferent fixed point for (2.3) and
+SI(x2) = SI(x1).
+3. If |c|p = α = β and |a − c2|p = α2, then x2 is indifferent fixed point for (2.3) and
+SI(x2) = Uα(x2),
+SI(x2) ∩ SI(x1) = ∅.
+4. If |c|p = α = β and |a − c2|p < α2, then x2 is attractive fixed point for (2.3) and
+A(x2) = Uα(x2) ⊂ Sα(0).
+5. If α < β, then x2 ∈ Sβ(0) is repelling fixed point for (2.3) and the inequality
+|f(x) − x2|p > |x − x2|p holds for all x ∈ Uβ(x2), x ̸= x2.
+Proof. 1. Let x ∈ Sr(x1), i.e., |x|p = r. Then, from the equalities (3.1), (3.2) and the
+properties of the p-adic norm, we have the following
+|f(x)|p =
+
+
+
+
+
+
+
+r,
+if r < α,
+≥ α,
+if α ≤ r ≤ β,
+|a|p
+r ,
+if r > β.
+From this equality, f(Sr(x1)) ⊂ Sr(x1) for arbitrary r < α, i.e. we have SI(x1) = Uα(0).
+
+ERGODICITY AND PERIODIC ORBITS
+5
+2. Note that |a|p = αβ. If |c|p < α = β, then |f ′(x2)|p =
+���1 − c2
+a
+���
+p = 1. From this x2 is
+indifferent fixed point. Let x ∈ Sr(x2), i.e., |x − x2|p = r. Then from the equality
+|f(x) − x2|p = |x − x2|p ·
+|x2(x − x2) + (x2
+2 − a)|p
+|(x − x2) + ˆx1|p|(x − x2) + ˆx2|p
+(3.3)
+we have |f(x) − x2|p = r for all r < α and |f(x) − x2|p ≥ r for r = α. Thus, f(Sr(x2)) ⊂
+Sr(x2) for arbitrary r < α, i.e. we have SI(x2) = Uα(x2). In this case, we have |x2|p =
+|c|p < α, so x2 ∈ Uα(0) = SI(x1). Since these two Siegel disks have the same radii and
+share a common point, they are the same, i.e., SI(x2) = SI(x1).
+3. If |c|p = α = β and |a − c2|p = α2, then |f ′(x2)|p =
+��� a−c2
+a
+���
+p = 1. From this x2 is
+indifferent fixed point. As above, from equation (3.3) we get SI(x2) = Uα(x2). However,
+in this case x2 ∈ Sα(0), so SI(x2) ∩ SI(x1) = ∅.
+4. If |c|p = α = β and |a − c2|p < α2, then |f ′(x2)|p =
+��� a−c2
+a
+���
+p < 1. From this x2 is
+attractive fixed point. Note that |x2|p = α. Let x ∈ Uα(x2) ⊂ Sα(0). Then from equality
+(3.3) and using the strong triangle inequality of the p-adic norm we derive the relation
+|f(x) − x2|p < |x − x2|p for all x ∈ Uα(x2). Similarly, if x /∈ Uα(x2), then we have the
+relation |f(x) − x2|p ≥ α.
+Note that, the set of valuations of p-adic norm is {pm| m ∈ Z}.
+Thus, the relation
+|f(x) − x2|p < |x − x2|p is equivalent to the relation |f(x) − x2|p ≤ 1
+p|x − x2|p. This means
+that the map f : Uα(x2) → Uα(x2) is a contraction map. According to the properties of
+contraction map, we have the equality A(x2) = Uα(x2).
+5. If α < β, then we have |x2|p = β, i.e., x2 ∈ Sβ(0). Also, |f ′(x2)|p =
+���1 − c2
+a
+���
+p = β
+α > 1.
+Let x ∈ Sr(x2), i.e., |x − x2|p = r. Then from the equality (3.3) we get
+|f(x) − x2|p =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+β
+α|x − x2|p,
+if r < α,
+≥ β,
+if r = α,
+β,
+if α < r < β,
+≤ β,
+if r = β,
+β,
+if r > β.
+From this we conclude that the inequality |f(x)−x2|p > |x−x2|p is holds for all x ∈ Uβ(x2),
+x ̸= x2.
+□
+Corollary 1. • The spheres Sr(x1) is invariant for f if and only if r < α.
+• The spheres Sr(x2) is invariant for f if and only if one of the statements holds
+a) |c|p < α = β and r < α;
+b) |c|p = α = β, |a − c2|p = α2 and r < α.
+
+6
+I.A. SATTAROV, E.T. ALIEV
+4. Ergodicity of the dynamical systems on invariant spheres
+Recall that an invariant measure is a measure that is preserved by some function. In
+ergodic theory of dynamical systems an invariant measure is very important .
+Let G be a topological group. If G is abelian and locally compact, then it is well known
+[7] that it has a nonzero translation-invariant measure µ, which is unique up to scalar. This
+is called the Haar measure.
+In the field of p-adic numbers let Σ be the minimal σ-algebra containing all open and
+closed (clopen) subsets.
+A measure µ(Vρ) = ρ, Vρ ∈ Σ is usually called a Haar measure, where Vρ is a ball with
+radius ρ.
+However, in some cases, the problem of studying the dynamical system of a function that
+mapping a compact subset of Qp to itself arises. At this time, is needed a measure defined
+on σ-algebra with the unit a compact set. If this compact set has some algebraic structure,
+then can we look at the natural Haar measure? If the considered compact set is a ball or a
+sphere, the answer to this question is positive, which is given as follows in [16].
+Let Vr(a) be the ball (Sr(a) be the sphere) with the center at the point a ∈ Qp and B is
+the algebra generated by clopen subsets of Vr(a) (Sr(a)). It is known that every element of
+B is a union of some balls Vρ(s) ⊂ Vr(a), s ∈ Vr(a) (Vρ(s) ⊂ Sr(a), s ∈ Sr(a)).
+Theorem 2. [16] A measure ¯µ : B → pZ is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ
+for all Vρ(s) ∈ B.
+Also, ergodic theory often deals with ergodic transformations. Here is the definition:
+Definition 1. [18] Let T : X → X be a measure-preserving transformation on a measure
+space (X, Σ, µ), with µ(X) = 1. Then T is ergodic if for every E in Σ with T −1(E) = E,
+either µ(E) = 0 or µ(E) = 1.
+In this section we are interested in ergodicity (with respect to Haar measure) of the
+dynamical systems on invariant spheres with the center at the fixed point..
+Remark 2. Corollary 1 in the previous section gives a classification of invariant spheres
+centered at a fixed point. Also, in part 2 of Theorem 1, it is proved that maximal Siegel discs
+consisting of union of invariant spheres fall on top of each other. Therefore, the center of
+invariant spheres is not significant when |c|p < α = β. However, when |c|p = α = β, it
+is necessary to consider separately the ergodicity of dynamical systems in invariant spheres
+with centers x1 and x2.
+For each invariant sphere we consider a measurable space (Sr(xi), B), here B is the algebra
+generated by closed subsets of Sr(xi), i = 1, 2. Every element of B is a union of some balls
+Vρ(s) ⊂ Sr(xi).
+A measure ¯µ : B → R is a Haar measure if it is defined by ¯µ(Vρ(s)) = ρ for all s ∈ Sr(xi)
+and ρ ∈ pZ such that Vρ(s) ⊂ Sr(xi).
+Note that Sr(xi) = Vr(xi) \ V r
+p (xi). So, we have ¯µ(Sr(xi)) = r(1 − 1
+p).
+
+ERGODICITY AND PERIODIC ORBITS
+7
+We consider normalized (probability) Haar measure:
+µ(Vρ(s)) = ¯µ(Vρ(s))
+¯µ(Sr(xi)) =
+pρ
+(p − 1)r.
+Theorem 3. Let Sr(xi), i = 1, 2 be invariant sphere for the function f given by (2.3).
+Then the function f : Sr(xi) → Sr(xi) is an isometry.
+Proof. By the Corollary 1, if the sphere Sr(xi), i = 1, 2 is invariant for (2.3), then r < α.
+Let i = 1. From relation x, y ∈ Sr(x1) we have |x|p = |y|p = r. Then, we get the following
+|f(x) − f(y)|p = |x − y|p ·
+|a|p|a − xy|p
+|(x − ˆx1)(x − ˆx2)(y − ˆx1)(y − ˆx2)|p
+.
+(4.1)
+Note that |a|p = αβ and |x|p = |y|p = r < α ≤ β. Then,
+|f(x) − f(y)|p = |x − y|p · α2β2
+α2β2 = |x − y|p.
+Consequently, the function f : Sr(x1) → Sr(x1) is an isometry.
+Let i = 2. Then by Corollary 1 we have two cases. If |c|p < α = β , then by Remark
+2, this case overlaps with case i = 1. If |c|p = α = β and |a − c2|p = α2, then by part 3
+of Theorem 1, we have the relation Sr(x2) ⊂ Sα(0) for all invariant sphere. So, we have
+|x − x2|p = r < α and |x|p = α for all x ∈ Sr(x2).
+Let x, y ∈ Sr(x2). Then
+|f(x) − f(y)|p = |x − y|p ·
+|a|p|(a − x2
+2) + x2(x2 − y) + y(x2 − x)|p
+|[(x − x2) + ˆx1][(x − x2) + ˆx2][(y − x2) + ˆx1][(y − x2) + ˆx2]|p
+.
+Note that |a|p = α2, |x − x2|p = |y − x2|p = r < α and |a − x2
+2|p = |a − c2|p = α2. Then,
+|f(x) − f(y)|p = |x − y|p · α4
+α4 = |x − y|p.
+Consequently, the function f : Sr(x2) → Sr(x2) is an isometry.
+□
+Corollary 2. Let the conditions of the above theorem be satisfied. Then f : Sr(xi) → Sr(xi),
+i = 1, 2 is a measure-preserving transformation on a measure space (Sr(xi), B, µ), where µ
+is a normalized Haar measure.
+In [16], given an important results about the dynamics of isometric maps, and since the
+function (2.3) we are considering is also an isometry, the results obtained in [16] are also
+relevant for the dynamics of the function (2.3), i.e., if Sr(xi), i = 1, 2 is invariant sphere for
+the function f given by (2.3), then we have the following:
+• The function f : Sr(xi) → Sr(xi), i = 1, 2 is bijection.
+• For any initial point x ∈ Sr(xi), i = 1, 2 (except fixed point) the orbit {f n(x)| n ∈ N}
+isn’t convergent.
+The result of the following Lemma is given as a condition in [16]. Let Sr(xi), i = 1, 2 be
+invariant sphere for the function f given by (2.3), then we denote ρ(r, x) = |f(x) − x|p for
+x ∈ Sr(xi).
+
+8
+I.A. SATTAROV, E.T. ALIEV
+Lemma 1. If r ̸= |c|p, then for the function f given by (2.3) the value ρ(r, x) does not
+depend to x.
+Proof. We consider all cases in Corollary 1. Let i = 1. Then r < α. By simple calculation
+we get
+ρ(r, x) =
+����
+ax
+x2 + cx + a − x
+����
+p
+= |x|2
+p ·
+|x + c|p
+|x − ˆx1|p|x − ˆx2|p
+=
+
+
+
+r2|c|p
+αβ ,
+if r < |c|p,
+r3
+αβ,
+if r > |c|p.
+Let i = 2. In this case, according to Remark 2, it is sufficient to prove the Lemma when
+|c|p = α = β. So, we have r = |x − x2|p = |x + c|p < α and
+ρ(r, x) =
+����
+ax
+x2 + cx + a − x
+����
+p
+= |x + c|p ·
+|(x + c) − c|2
+p
+|(x + c) + ˆx1|p|(x + c) + ˆx2|p
+= r.
+□
+So, we denote ρ(r) = |f(x) − x|p for all x ∈ Sr(xi), i = 1, 2, r ̸= |c|p. In that case, we
+have the following assertions from [16]:
+• The ball with radius ρ(r) is minimal invariant ball for f : Sr(xi) → Sr(xi), i = 1, 2,
+r ̸= |c|p.
+• Let µ be normalized Haar measure on Sr(xi). Then
+a) the dynamical system (Sr(xi), f, µ) is not ergodic for all p ≥ 3;
+b) the dynamical system (Sr(xi), f, µ) may be ergodic if and only if r = 2ρ(r) for
+p = 2.
+Let p = 2.
+Then according to the above the dynamical system (Sr(x2), f, µ) is not
+ergodic, because r = ρ(r) for i = 2.
+If i = 1, then x1 = 0 and we consider the dynamical system (Sr(0), f, µ).
+Recall Z2 = {x ∈ Q2 : |x|2 ≤ 1}. So we have 1 + 2Z2 = S1(0). The following theorem
+gives a criterion of ergodicity for the rational functions mapping S1(0) to itself:
+Theorem 4. [9] Let f, g : 1 + 2Z2 → 1 + 2Z2 be polynomials whose coefficients are 2-adic
+integers.
+Set f(x) = �
+i aixi, g(x) = �
+i bixi, and
+A1 =
+�
+i odd
+ai,
+A2 =
+�
+i even
+ai,
+B1 =
+�
+i odd
+bi,
+B2 =
+�
+i even
+bi.
+The rational function R =
+f
+g is ergodic if and only if one of the following situations
+occurs:
+(1) A1 = 1(mod 4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 1(mod 4).
+(2) A1 = 3(mod4), A2 = 2(mod 4), B1 = 0(mod 4) and B2 = 3(mod 4).
+(3) A1 = 1(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 1(mod 4).
+(4) A1 = 3(mod 4), A2 = 0(mod 4), B1 = 2(mod 4) and B2 = 3(mod 4).
+(5) One of the previous cases with f and g interchanged.
+
+ERGODICITY AND PERIODIC ORBITS
+9
+Consider x = g(t) = r−1t for t ∈ S1(0), then g−1 ◦ f ◦ g : S1(0) → S1(0). Let B (resp.
+B1) be the algebra generated by closed subsets of Sr(0) (resp. S1(0)), and µ (resp. µ1) be
+normalized Haar measure on B (resp. B1).
+Theorem 5. [14] The dynamical system (Sr(0), f, µ) is ergodic if and only if
+(S1(0), g−1 ◦ f ◦ g, µ1) is ergodic.
+Now using the above mentioned results for (2.3) when p = 2 and we prove the following
+Theorem 6. Let p = 2. Then the dynamical system (Sr(0), f, µ) is ergodic if and only if
+|c|2 = β and r = α
+2 .
+Proof. Let r = 2l, α = 2m, β = 2k and |c|2 = 2q. Since α ≤ β we have m ≤ k. Also, since
+c = −ˆx1 − ˆx2 and a = ˆx1ˆx2 we have q ≤ k and |a|2 = 2m+k.
+Note that the sphere S2l(0) is invariant for f iff l < m.
+We consider the function
+g : S1(0) → Sr(0) defined by x = g(t) = 2−lt. Note that the function
+g−1(f(g(t))) : S1(0) → S1(0) has the following form
+g−1(f(g(t))) =
+t
+2−2l
+a t2 + 2−lc
+a t + 1
+,
+(4.2)
+for the function f given by (2.3). Note that k, l, m, q ∈ Z, l < m ≤ k and q ≤ k. So we
+have the inequalities l − m ≤ −1 and l − k ≤ −1. In (4.2) we can easily see the following
+����
+2−2l
+a t2
+����
+2
+= 22l−(m+k) ≤ 2−2,
+����
+2−lc
+a t
+����
+2
+= 2l+q−(m+k) ≤ 2−1.
+Consequently,
+t =: γ1(t),
+is such that γ1 : 1 + 2Z2 → 1 + 2Z2
+and
+2−2l
+a t2 + 2−lc
+a t + 1 =: γ2(t) is such that γ2 : 1 + 2Z2 → 1 + 2Z2.
+Hence the function (4.2) satisfies all condition of Theorem 4, therefore using this theorem,
+we get
+A1 = 1,
+A2 = 0,
+B1 = 2−lc
+a
+and B2 = 2−2l
+a
++ 1.
+Moreover,
+A1 = 1(mod 4),
+A2 = 0(mod 4),
+B1 ∈ 2m+k−(l+q)(1 + 2Z2) and B2 = 1(mod 4).
+By these relations and Theorem 4 we get m+k−(l+q) = (m−l)+(k−q) = 1. Note that
+l < m and q ≤ k. Therefore we conclude that the dynamical system (S1(0), g−1 ◦ f ◦ g, µ1)
+is ergodic if and only if q = k and l = m − 1, i.e., |c|2 = β and r = α
+2 . Consequently, by
+Theorem 5, (Sr(0), f, µ) is ergodic if and only if |c|2 = β and r = α
+2 .
+□
+
+10
+I.A. SATTAROV, E.T. ALIEV
+5. Periodic orbits
+In this section we are interested in periodic trajectories and their characteristics. Since
+our function is an isometry on an invariant sphere, we get the following result about periodic
+trajectories from [16]:
+Theorem 7. If the dynamical system (Sr(xi), f), i = 1, 2 has n-periodic orbit
+y0 → y1 → ... → yn → y0,
+then the following statements hold:
+1. yk ∈ Vρ(r)(y0) for all k ∈ {1, 2, ..., n};
+2. Character of periodic points is indifferent;
+3. If ρ ≤ ρ(r), then we have f(Sρ(yk)) ⊂ Sρ(yk+1) for any k ∈ {0, 1, ...n − 1} and
+f(Sρ(yn)) ⊂ Sρ(y0).
+Now we prove the following theorems about the existence of 2-periodic and 3-periodic
+trajectories:
+Theorem 8. If
+√
+c2 − 2a ∈ Qp, then the function (2.3) has unique 2-periodic orbit {t1, t2},
+where t1,2 = −c ±
+√
+c2 − 2a.
+Proof. We consider the equation
+f 2(x) − x
+f(x) − x = 0.
+Then we obtain the following
+(x2 + 2cx + 2a)(x2 + cx + a) = 0.
+Since x2 + cx + a ̸= 0, we get x2 + 2cx + 2a = 0, and t1,2 = −c ±
+√
+c2 − 2a.
+□
+Theorem 9. Let Sr(xi), i = 1, 2 be invariant sphere for (2.3) and assume that the param-
+eter a ∈ Sr(xi). Then the function (2.3) has 3-periodic orbit
+�
+a, f(a), f 2(a)
+�
+if and only
+if
+(a, c) ∈
+�
+(h(q), qh(q) − 1) : q ∈ Qp \
+�
+0, −1, −2
+3
+�
+, |h(q)|p = r
+�
+,
+for i = 1,
+(5.1)
+(a, c) ∈
+�
+(h(q), qh(q) − 1) : q ∈ Qp \
+�
+0, −1, −2
+3
+�
+, |h(q)(q + 1) − 1|p = r
+�
+,
+for i = 2,
+(5.2)
+where h(q) =
+3q2+2q
+6q3+11q2+6q+1.
+Proof. We consider the equation
+f 3(x) − x
+f(x) − x = 0.
+By simplifying this equation, we get the following equation
+P(x) = x6 + 6cx5 + (11c2 + 6a)x4 + (6c3 + 20ac)x3 + (15ac2 + 9a2)x2 + 12a2cx + 3a3 = 0.
+
+ERGODICITY AND PERIODIC ORBITS
+11
+Necessity. Let a ∈ Sr(xi) be a 3-periodic point. Then P(a) = 0 and from this we have the
+equality
+a3 + 6(c + 1)a2 + (11c + 9)(c + 1)a + 3(2c + 1)(c + 1)2 = 0.
+(5.3)
+According to equality (5.3), since a ̸= 0, we have c ̸= −1. Denote
+q = c + 1
+a
+.
+Then by (5.3) we get (6q3 + 11q2 + 6q + 1)a − (3q2 + 2q) = 0.
+If we denote
+a := h(q) =
+3q2 + 2q
+6q3 + 11q2 + 6q + 1,
+then c = qh(q) − 1. Notice that h(q) is undefined at q = −1. Applying the conditions that
+a(c + 1) ̸= 0 we see that q ̸= 0 and q ̸= − 2
+3.
+For i = 1, we have |a|p = |h(q)|p = r, analogically for i = 2 we have
+|a + c|p = |h(q)(q + 1) − 1|p = r. Summarizing the above, we get (5.1) and (5.2).
+Sufficiency.
+Let conditions (5.1) and (5.2) be satisfied.
+Then it is easy to see that
+P(a) = 0. Hence, a ∈ Sr(xi) is 3-periodic point for f given by (2.3).
+□
+6. Availability of data
+The datasets supporting the conclusions of this article are included in the article.
+Acknowledgements
+We thank our supervisor U.A. Rozikov for the useful discussions.
+References
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+Appl. 398(2) (2013), 553–566.
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+12
+I.A. SATTAROV, E.T. ALIEV
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+(2012), 376–383.
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+Fixed Points, Results in Mathematics, 100(75) (2020), 1–37.
+[14] U.A. Rozikov, I.A. Sattarov. p-adic dynamical systems of (2, 2)-rational functions with unique fixed
+point. Chaos, Solitons and Fractals, 105 (2017), 260–270.
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+ax
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+Ultrametric Analysis and Applications, 11(1) (2019), 77–87.
+[16] I.A. Sattarov. Group structure of the p-adic ball and dynamical system of isometry on a sphere.
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+Applications, 7(1) (2015), 39–55.
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+I. A. Sattarov, Namangan Satate University, 316, Uychi str., 160100, Namangan, Uzbekistan.
+Email address: [email protected]
+E. T. Aliev, Namangan Institute of Engineering Technology, 7, Kosonsoy str., 160115,
+Namangan, Uzbekistan.
+Email address: [email protected]
+

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+Enhancing the Accuracy of Density Functional Tight Binding Models Through
+ChIMES Many-body Interaction Potentials
+Nir Goldman,1, 2 Laurence E. Fried,1 Rebecca K. Lindsey,3 C. Huy Pham,1 and R.
+Dettori1
+1)Physical and Life Sciences Directorate, Lawrence Livermore National Laboratory,
+Livermore, CA 94550 USAa)
+2)Department of Chemical Engineering, University of California, Davis, California 95616,
+United States
+3)Department of Chemical Engineering, University of Michigan, Ann Arbor,
+Michigan 48109, United States
+(Dated: 5 January 2023)
+Semi-empirical quantum models such as Density Functional Tight Binding (DFTB) are
+attractive methods for obtaining quantum simulation data at longer time and length scale
+than possible with standard approaches. However, application of these models can require
+lengthy effort due to the lack of a systematic approach for their development. In this work,
+we discuss use of the Chebyshev Interaction Model for Efficient Simulation (ChIMES) to
+create rapidly parameterized DFTB models which exhibit strong transferability due to the
+inclusion of many-body interactions that might otherwise be underestimated. We apply
+our modeling approach to silicon polymorphs and review previous work on titanium hy-
+dride. We also review creation of a general purpose DFTB/ChIMES model for organic
+molecules and compounds that approaches hybrid functional and coupled cluster accuracy
+with two orders of magnitude fewer parameters than similar neural network approaches.
+In all cases, DFTB/ChIMES yields similar accuracy to the underlying quantum method
+with orders of magnitude improvement in computational cost. Our developments provide
+a way to create computationally efficient and highly accurate semi-empirical models for
+studies where physical and chemical properties can be difficult to interrogate directly and
+there is historically a significant reliance on theoretical approaches for interpretation and
+validation of experimental results.
+a)Electronic mail: [email protected]
+1
+arXiv:2301.01733v1  [cond-mat.mtrl-sci]  4 Jan 2023
+
+I.
+INTRODUCTION
+Atomistic calculation approaches for materials modeling can be used as an independent route
+to aid in new materials synthesis1, characterizing mixtures for use as fuel2,3, or quantifying rates
+for chemical decomposition of organic materials4. These types of studies generally rely on quan-
+tum mechanical approaches such as Kohn-Sham Density Functional Theory (DFT) in order to aid
+in experimental interpretation and/or new materials design. In particular, DFT has been shown ex-
+tensively to yield accurate descriptions of condensed phase physical and chemical data, such as the
+material equation of state under compressive or tensile loads5, heats of formation/mixture of new
+phases6,7, and the energetics of chemical bond breaking and forming under reactive conditions8.
+However, standard DFT is also renown for its significant computational expense and poor com-
+putational scaling (generally O(N 3)) resulting from solving for the Kohn-Sham eigenstates. As a
+result, DFT molecular dynamics (MD) simulations can be limited to system sizes of hundreds of
+atoms for timescales of tens of picoseconds or smaller for many systems9. In contrast, many pro-
+cesses of interest have properties that can span orders of magnitude larger scales, including large-
+scale carbon heterocycle synthesis10, the rational design of 3D materials11, and defect formation
+and grain boundary interactions in crystalline systems12. Thus, the need for alternate simulation
+approaches remains a highly active research area where the goal is to develop methods that can
+harness the accuracy of DFT while yielding vastly improved computational efficiency and scaling.
+In this regard, machine learning approaches for the development of interatomic atomic po-
+tentials have been an effective route for modeling materials under reactive and nonreactive
+conditions13,14. For example, neural networks have been used successfully to model structural
+properties of catalytic materials15 as well as the phase stability of high-entropy ceramics16. Gaus-
+sian Process Regression in the form of the Gaussian Approximation Potential (GAP) has been
+used for a number of materials, including silicon based materials17. Regardless, the development
+of these potentials tends to remain a highly labor-intensive task, where frequently a high-degree
+of expertise and months to years of human effort are required for a single application area. As
+a result, it can be difficult for these efforts to keep up with experimental needs particularly in
+the area of materials synthesis, where the number of permutations of different starting materials,
+thermodynamic conditions, and catalysts can be combinatorially large.
+Semi-empirical quantum mechanical approaches hold promise as a middle ground for acceler-
+ated simulations with a high degree of accuracy. These methods combine approximate quantum
+2
+
+mechanics with empirical functions to yield approaches that can achieve several orders of magni-
+tude longer time scales in quantum MD simulations.18,19 In addition, semi-empirical approaches
+tend to utilize significantly fewer computational resources, allowing for ensembles of statistically
+independent trajectories and improved statistical sampling of desired properties.20 These methods
+also tend to show much stronger transferability to systems and conditions outside of their training
+set compared to interatomic potentials, in part due to the accuracy of the approximate quantum
+mechanics and subsequent reduced reliance on empirical functions.21
+Density Functional Tight Binding (DFTB) is one such semi-empirical quantum mechanical
+method22,23 that has had widespread success in modeling both gas-phase molecules24 as well
+as condensed matter under inert and reactive conditions25–27, including extreme pressures and
+temperatures28,29. The DFTB total energy is derived from an expansion of the Kohn-Sham en-
+ergy to either second or third-order in charge fluctuations, resulting in the following expression:
+EDFTB = EBS + ECoul + Erep.
+(1)
+Here, EBS corresponds to the band structure energy, ECoul is the charge fluctuation term, and
+Erep is the repulsive energy. EBS is calculated as a sum over occupied electronic states from the
+DFTB Hamiltonian. The DFTB Hamiltonian matrix elements are determined from pre-tabulated
+Slater-Koster tables derived from reference calculations with a minimal basis set. The onsite
+matrix elements are the free-atom orbital energies and the off-site terms are computed with a two-
+center approximation where both wavefunctions and electron density are subjected to confining
+potentials. Erep corresponds to ion-ion repulsions, as well as Hartree and exchange-correlation
+double counting terms. This term can be expressed as an empirical function where parameters
+are fit to reproduce high-level quantum or experimental reference data. In practice, an additional
+dispersion correction can be included, including those in standard use for DFT calculations30,31.
+DFTB is approximately three orders of magnitude more efficient than DFT calculations though
+it also tends to exhibit O(N 3) scaling due to the need to solve for the band structure eigenstates.
+DFTB has been shown to exhibit transferability across element types and diverse conditions32–34
+and has been applied to a broad range of materials35–39.
+However, DFTB model development can be challenging in terms of optimizing the hyperpa-
+rameters needed for the approximate quantum mechanical parts of the calculations. These include
+the separate confining potentials for the wavefunctions and electron density (which can be differ-
+ent for each angular momentum channel of an element),38 choice of second-order vs. third-order
+3
+
+charge fluctuations for the energy expression40, and whether to use density or potential superposi-
+tion when computing the Slater-Koster tables.36,41 The DFTB Hamiltonian tends to be highly sen-
+sitive to these options42, and in general there does not exist a predefined recipe for how to choose
+these parameters nor how to explore that specific phase space. Prediction of physical and/or chem-
+ical properties are in turn are closely coupled to the empirical repulsive energy, which itself has a
+wide variety of options in terms of functional form and data to be fit35. ERep is usually taken to be
+strictly pairwise (two-center), though a number of systems can require many-body terms as well
+for accurate predictions28. Novel approaches for determination of ERep include constrained spline
+optimization34, neural networks43,44, and Gaussian Process Regression45,46. Machine learning ap-
+proaches though tend to be highly data intensive14 and prone to overfitting21, which can pose dif-
+ficulties for any method that leverages these techniques. Thus, DFTB method development would
+be holistically improved through a more automatic method for parameterization, where candidate
+models could be screened rapidly and efficiently, thereby allowing the user to quickly determine
+an optimal model for their specific needs.
+In this work, we discuss our recent efforts to overcome these issues through use of the Cheby-
+shev Interaction Model for Efficient simulation (ChIMES),47,48 which can be used to determine
+ERep for molecular and condensed phase systems relatively quickly and with comparatively lower
+data requirements. ChIMES is a many-body reactive force field based on linear combinations
+of Chebyshev polynomials. It was initially developed for pure MD simulation (i.e., where all
+aspects of a quantum mechanical calculation have been mapped onto the ChIMES functional
+form). This has included both non-reactive and reactive materials, such as water under ambi-
+ent and high pressure-temperature conditions49,50, high pressure C/O systems51,52, and detonating
+energetic materials53. DFTB/ChIMES models have been created for a wide variety of materials,
+including actinides and their oxides54,55, titanium-based systems36, and silicon (discussed below).
+Additionally, ChIMES has been used to improve the accuracy of DFTB by including many-body
+energies and forces through ∆-learning, where ChIMES augments a pre-existing DFTB param-
+eterization for organic materials under ambient56 and reactive conditions39. We note that similar
+to other machine-learning methods21, ChIMES can be used within any semi-empirical quantum
+mechanical approach. However, we choose to focus on DFTB due to its close resemblance to
+Kohn-Sham DFT as well as its proven accuracy for a variety of materials and conditions.
+We begin with a brief discussion of the ChIMES formalism, including discussion of its func-
+tional form and methods for optimization. Next we present some recent results on a general pur-
+4
+
+pose DFTB/ChIMES model for silicon polymorphs, which has remained an outstanding issue in
+DFTB model development. We note that all DFTB calculations discussed within this work were
+performed with the DFTB+ code57,58. We then summarize previous work on a semi-automated
+workflow for screening DFTB hyperparameters and ERep determination in creating a models for
+TiH2, a candidate hydrogen storage material with several potential uses. Finally, we review our
+recent results in using ChIMES to create DFTB models that approach hybrid-functional and cou-
+pled cluster accuracy for organic compounds and molecular solids. In all cases, the advantages to
+use of DFTB/ChIMES lies in its rapid parameterization time, small data requirements relative to
+other machine-learned approaches, and the relative ease with which overfitting can be prevented
+due to regularization within linear optimization approaches as well as the orthogonal nature of the
+underlying basis set.
+II.
+METHODS
+A.
+ChIMES Formalism
+The design philosophy behind ChIMES is based on a many-body expansion of the DFT total
+energy. Briefly, the DFT total energy can be thought of as a sum of contributions of clusters
+containing different numbers of atoms:
+EDFT =
+na
+�
+i1
+1Ei1+
+na
+�
+i1>i2
+2Ei1i2+
+na
+�
+i1>i2>i3
+3Ei1i2i3+
+na
+�
+i1>i2>i3>i4
+4Ei1i2i3i4+· · ·+
+na
+�
+i1>i2... inB−1>inB
+nBEi1i2...inB.
+(2)
+Here, the one-body energies, 1Ei1, correspond to the atomic energy constants, the two-body ener-
+gies, 2Ei1i2, to all pair-wise energies with indices {i1, i2}, the three-body energies, 3Ei1i2i3, to all
+triplet energies with indicies {i1, i2, i3}, etc., all the way up to some predeterimed maximum bod-
+iedness, nB. These terms are summed over all cluster combinations within the system containing
+na total number of atoms.
+In the ChIMES formalism, we represent each of the terms in our n-body expansion as a lin-
+ear combination of Chebyshev polynomials. Chebyshev polynomials of the first kind of order m
+are defined by the expression Tm (cos θ) = cos (mθ), more commonly written as Tm(x), where
+x = cos θ and thus exists over the range [−1, 1]. Chebyshev polynomials offer a number of dis-
+tinct advantages for interpolation that bear mentioning. Chebyshev polynomials of the first kind
+5
+
+are orthogonal with respect to the weighting function 1/
+√
+1 − x2. They can be computed with
+a recurrence relationship and define a complete basis set, allowing for arbitrary complexity in a
+potential energy surface. Their orthogonality allows for simple regularization where higher-order
+polynomial coefficients can be set to zero without necessarily adversely affecting the quality of
+the optimization. Polynomial expansions with Chebyshev polynomials of the first kind will have
+exponentially decreasing coefficients for higher-order terms due to their monic form, helping to
+prevent overfitting. In addition, they yield a “nearly optimal” error function, where the error in
+an expansion will closely resemble a minimax polynomial. The derivaties of Chebyshev polyno-
+mials of the first kind are related to Chebyshev polynomials of the kind Um(x) by the expression
+dTm/dx = mUm−1, where Um (cos θ) = sin [(n + 1) θ] /sin θ. Chebyshev polynomials of the
+second kind also form an orthogonal basis set (with respect to the weighting function
+√
+1 − x2)
+and can also be generated via a recurrence relation. This can allow for arbitrary complexity for
+structural optimization or molecular dynamics calculations, where atomic forces are needed.
+As a result, we can now write the two-body (2B) energy term in Equation 2 as the following
+expression:
+2Ei1i2 = fp (ri1i2) + f
+ei1ei2
+c
+(ri1i2)
+O2
+�
+m=1
+C
+ei1ei2
+m
+Tm(s
+ei1ei2
+i1i2 )
+(3)
+In this case, C
+ei1ei2
+m
+is the corresponding permutationally invariant coefficient for the interaction
+between atom types ei1 and ei2, taken from the set of all possible element types, {e}. Tm
+�
+s
+ei1ei2
+i1i2
+�
+represents a Chebyshev polynomial of order m, and s
+ei1ei2
+i1i2
+is the pair distance transformed to
+occur over the interval [−1, 1] using a Morse-like function59,60. For that coordinate transform,
+s
+ei1ei2
+i1i2
+∝ exp (−ri1i2/λe1e2) and λe1e2 is an element-pair distance scaling constant, usually taken
+to be the peak position of the first coordination shell. Further details are discussed in Ref. 47. The
+term f
+ei1ei2
+c
+(ri1i2) is a Tersoff cutoff function61 which is set to zero beyond a maximum distance
+defined for a given {e1, e2} pair set. In order to prevent sampling of ri1i2 distances below what is
+sampled in our DFT training set, we introduce use of a smooth penalty function fp(ri1i2).
+We can create greater than two-body orthogonal polynomials by defining a cluster of size n and
+taking the product of the Chebyshev polynomials derived from the constituent
+�n
+2
+�
+unique pairs.
+For example, the three-body polynomials will be products of
+�3
+2
+�
+= 3 two-body polynomials. We
+thus write the ChIMES three-body (3B) energy as the following:
+6
+
+3Ei1i2i3 = f
+ei1ei2
+c
+(ri1i2) f
+ei1ei3
+c
+(ri1i3) f
+ei2ei3
+c
+(ri2i3)
+O3
+�
+m=0
+O3
+�
+p=0
+O3
+�
+q=0
+′
+C
+ei1ei2ei3
+mpq
+Tm
+�
+s
+ei1ei2
+i1i2
+�
+Tp
+�
+s
+ei1ei3
+i1i3
+�
+Tq
+�
+s
+ei2ei3
+i2i3
+�
+.
+(4)
+We take a triple sum for the i1i2, i1i3, and i2i3 polynomials over the hypercube up to O3, and
+include a single permutationally invariant coefficient for each set of powers and atom types,
+C
+ei1ei2ei3
+mpq
+. We use the primed sum to denote that only terms for which two or more of the m, p, q
+polynomial powers are greater than zero are included in order to guarantee that three distinct
+atom-centers are evaluated. The expression for 3Ei1i2i3 also contains the fc smoothly varying cut-
+off functions for each constituent pair distance. Penalty functions are not included in this case and
+instead are handled entirely by the two-body interaction.
+Higher bodied terms are included in ChIMES in a similar fashion. For example, four-body (4B)
+terms are regularly included in ChIMES optimizations53, where 4Ei1i2i3i4 is now determined from
+the sum over the product of the
+�4
+2
+�
+= 6 constituent pair-wise polynomials multiplied by a single
+permutationally invariant coefficient. In practice, even higher bodied terms could be included in
+ChIMES, though this can lead to a combinatorially large polynomial space and hence parameter
+explosion that can lead to overfitting and excessive computational expense. Hence, the norm
+with ChIMES optimization is generally to include up to four-body terms, though DFTB/ChIMES
+models tend to be converged with up to three-body terms, only.36,39,54–56
+Optimal ChIMES parameters (the coefficients of linear combination) can then readily be deter-
+mined through the overdetermined matrix equation wAC = wBrep. The matrix A corresponds
+to the derivatives of the ChIMES energy or force expression with respect to the fitting coefficients.
+The column vectors C and Brep correspond to the linear ChIMES coefficients for which we are
+solving and the numerical values for the training data, respectively. The symbol w corresponds
+to a diagonal matrix of weights to be applied to the elements of Brep and rows of A. This linear
+least-squares optimization problem can be solved for with any number of established algorithms,
+discussed below.
+B.
+ChIMES optimization for ERep or ∆-learning
+The ChIMES training set for determination of ERep or ∆-learning proceed in a similar fashion.
+ERep training is computed by calculating DFTB forces (F), stress tensor components (σ), and
+7
+
+possibly system energies Etot for each configuration in the training set with the chosen set of
+Hamiltonian parameters (i.e., {Rψ}, {Rn}, density or potential superposition, second or third-
+order DFTB) with zero values for those components from ERep. These “repulsive energy free”
+results are then subtracted from the DFT values for those quantities, i.e.,
+Eτ∗
+Rep = Eτ
+DFTi − Eτ
+QM,DFTBi
+F τ∗
+Repαi = F τ
+DFTαi − F τ
+QM,DFTBαi
+στ∗
+Repαβ = στ
+DFTαβ − στ
+QM,DFTBαβ
+(5)
+Here, τ corresponds to a specific MD configuration, α and β to the cartesian directions, and i is
+the atomic index. In practice, we have used the diagonal components of the stress tensor, only
+(i.e., α = β in Equation 5). The ‘*’ is used to denote that the quantities being computed are part of
+the training set, and ‘QM,DFTB’ refers to the quantum components of the DFTB calculation, i.e.,
+only forces and stresses from EBS and ECoul. Calculation of a ∆-learning training set is identical
+with the exception that the quantities in Equation 5 are no longer repulsive energy free but instead
+contain terms from the DFTB repulsive energy model of choice. This results in the following
+objective function:
+Fobj =
+�
+�
+�
+� 1
+Nd
+×
+� M
+�
+τ=1
+N
+�
+i=1
+3
+�
+α=1
+w2
+Fαi (∆Fαi)2 +
+M
+�
+τ=1
+3
+�
+α=1
+w2σαα (∆σαα)2 +
+M
+�
+τ=1
+w2
+Ei (∆Ei)2
+�
+,
+(6)
+where M is the total number of configurations in the training set and Nd is the total number of data
+entries (3MN force components plus 3M stress tensor components plus M energy components).
+In addition, ∆Fαi = F τ
+ChIMESαi − F τ∗
+Repαi, ∆σαβ = στ
+ChIMESαβ − στ∗
+Repαβ, and ∆Ei = Eτ
+ChIMESi −
+Eτ∗
+Repi.
+ChIMES bears some resemblance to the Atomic Cluster Expansion approach (ACE)62,63, where
+many-body interactions are represented by a product of Chebyshev polynomials and real spherical
+harmonics. These models also differ from ChIMES in that the underlying polynomial basis set is
+atom-centered (similar in spirit to an embedded atom model64) rather than using a cluster approach
+as we adopt here. Similarly, the spectral neighbor analysis potential (SNAP) uses bispectrum
+components to compute the total energy of a system as a sum over atom energies, which are
+expressed as a weighted sum over bispectrum components65.
+8
+
+C.
+Linear least-squares approaches for ChIMES optimization
+The ChIMES potential is linear with respect to the fitting coefficients, which allows for use
+of powerful global optimization tools that are unavailable to non-linear machine-learned models.
+In our efforts, we have focussed on the Singular Value Decomposition (SVD) and Least-Angle
+Regression (LARS) with Least Absolute Selection and Shrinkage Operator (LASSO) regulariza-
+tion methods. We now offer a brief discussion of each method and leave details to the pertinent
+references.
+SVD66 solves for optimal fitting coefficients directly by performing an eigendecomposition
+of the generally rectangular A matrix and computing its pseudo-inverse. Regularization can be
+performed by setting singular values (eigenvalues of the square matrix in the SVD decomposition)
+with an absolute value below a given threshold to zero. In our work, we take this parameter to be
+Dmaxϵ, where Dmax is the maximum singular value of A and ϵ is a factor below a value of one.
+LARS is a type of forward step-wise or iterative regression67,68. Here, all model coefficients are
+initialized to zero and the covariate (i.e., polynomial values) most correlated to the error residual is
+determined (i.e., those having the most significant impact on the fit). The corresponding ChIMES
+parameter is modified incrementally to minimize the error residual until a second covariate yields
+an equal correlation. At this point, it is included in the active parameter set and both coefficients are
+modified simultaneously. The process continues until all coefficients are included in the solution,
+at which point a result equivalent to ordinary least squares fitting is obtained. In practice, LARS
+optimization can be performed using only a subset of all possible parameters.
+LASSO69 is an L1-norm regularization method whereby regularization is based on the sum
+of the absolute values of the fitting coefficients, which has the effect of shrinking a subset of
+parameters to zero. In this case, the objective function Fobj (Equation 6) is minimized with the
+following additional constraint:
+F LASSO
+obj
+= Fobj + 2α
+ni
+�
+i=1
+|ci| .
+(7)
+Here, ni is the total number of unique fitting parameters, ci. The parameter α regularizes the
+magnitude of the fitting coefficients, which reduces possible overfitting. The LASSO method
+can be implemented as a variant of LARS where parameters are either added or removed at each
+solution stage. We find the LASSO variant of LARS to be numerically stable for ill-conditioned
+A matrices, which are often found in force matching.
+9
+
+III.
+RESULTS
+A.
+DFTB/ChIMES Models for Silicon Polymorphs
+Silicon has proven to be a significant challenge for DFTB model parameterization likely due to
+the fact that its different polymorphs can have different coordination numbers and nearest neigh-
+bor distances. This yields a variety of bond lengths and energies that need to be accounted for in
+order to obtain a single, transferable DFTB model that does not have to be specific for a given solid
+phase. Previous work has shown that standard two-body repulsive energies do not exhibit sufficient
+complexity to accurately account for several Si phases with different bonding environments,34 in
+contrast to carbon, where multiple phases can be represented by a single two-body polynomial
+expansion70. Neural network (NN) approaches have been used for the repulsive energy in order
+to account for many-body interactions in ERep,44 and the results are promising. NN approaches
+though generally require large amounts of data and can frequently optimize to local minima, po-
+tentially complicating their use. Here, we attempt to overcome this issue by creating a many-body
+ChIMES ERep for silicon that is transferable to a number of different Si polymorphs as well as
+prediction of vibrational spectra and calculation of defect formation energies.
+In our work, we target two previous Si DFTB parameterizations, pbc-0-371 and siband-1-1,41
+which have different strengths and weaknesses. The pbc-0-3 parameter was creating using density
+superposition (i.e., the quantum mechanical potential VQM (ρ) was expressed as V (ρA + ρB) for
+atoms A and B) , which tends to be preferred due to its improved representation of chemical
+bonding and vibrations36. However, d-orbital interactions were not tabulated aside from the d-
+orbital onsite energy, which could have ramifications for some material properties. In contrast,
+the siband-1-1 parameter set was specifically created with d-orbital interactions but with potential
+superposition (i.e., VQM (ρ) = V (ρA) + V (ρB)) in order to yield accurate prediction of electronic
+properties, including the electronic band structure of Si-containing solids. In addition, the siband-
+1-1 parameter set does not contain a repulsive energy of any sort, precluding its use in structural
+relaxation or MD simulation which severely limits its usefulness overall.
+Our goal is to thus to create new ChIMES ERep potentials for each set of Slater-Koster interac-
+tion parameters using identical DFT training data and ChIMES hyperparameters in order to com-
+pare and contrast the effectiveness of each as a possible one-fits-all model. Calculations for our sil-
+icon DFT dataset were performed using the Vienna ab initio Simulation Package (VASP)72–74, with
+10
+
+projector-augmented wave function (PAW) pseudopotentials75,76 and the Perdew-Burke-Ernzerhof
+exchange-correlation functional (PBE)77. We found our results to be converged with a planewave
+cutoff of 500 eV, which was used in all of the calculations discussed here. We have used an
+electron density convergence criteria of 10−6 eV, with a force convergence of 10−2 eV/ ˚A for all
+geometry/cell optimizations. The Mermin functional78 smearing was set to 0.03 eV for all calcu-
+lations performed in this work. The system energy and pressure was found to be converged with
+sampling of the Brillouin Zone with a 2 × 2 × 2 Monkhorst-Pack mesh79 for all supercells. We
+then generated cold curves for each phase by isotropically expanding and contracting the simula-
+tion cell lattice. Here, we used a diamond structure supercell of 64 atoms, a bcc structure of 54
+atoms, a simple cubic structure of 64 atoms, and a graphene sheet of 32 atoms. This yielded an
+initial set of 463 configurations for our ChIMES ERep optimization.
+In order to sample forces from a variety different configurations, we have also included MD
+data for the diamond and graphene phases, using the same number of atoms in each supercell as
+before. These supercells were isotropically expanded and contracted between 90% to 110% of
+the ground-state density. Each MD simulation was run for ∼5 picoseconds at 600 K, from which
+we took snapshots at fixed intervals of ∼200 femtoseconds for our training set. This yielded
+an additional 405 configurations for our ChIMES ERep determination. In all, our final training set
+contained a total of 838 configurations of different silicon phases. ChIMES ERep optimization was
+then performed using values of rmin = 2.0 ˚A and rmax = 4.0 ˚A. The value of rmax was informed
+in part from previous development of a neural network repulsive energy,34 which resulted in a
+minimization of the root mean square (RMS) error in our fit. In addition, we found that a value of
+rmax = 4.0 ˚A yielded an improved description of the expanded states in our training set, where the
+bonded interactions between Si atoms is longer than the ground-state.
+We now refer to our ChIMES model based on pbc-0-3 as pbc/ChIMES and our model based
+on siband-1-1 as siband/ChIMES. Both pbc/ChIMES and siband/ChIMES were created with a 2B
+order of 12, 3B order of 8, and a LASSO regularization parameter (α) value of 10−3, similar to
+previous efforts36. We have used the Morse coordinate transform with a value of λ = 2.4 ˚A, which
+corresponds to the first peak in the diamond phase radial distribution function. For pbc/ChIMES,
+this yielded an overall RMS error of 1.44 eV/ ˚A in the forces, 0.43 GPA in the pressure, and
+0.038 eV/atom in energy. The RMS errors for siband/ChIMES were slightly higher, with values
+of 2.22 eV/ ˚A for the forces, 0.55 GPa for the pressure, and 0.16 eV/atom for the energy. Use
+of a Chebyshev basis set 2B order of 16, 3B order of 12, and 4B order of 4 yielded reduction in
+11
+
+the RMS errors of < 1% with similarly marginal improvement in validation quantities such as the
+computed defect energies. Use of a value of λ = 3.0 ˚A also had only a small effect on the resulting
+model. All ChIMES/DFTB calculations were performed with self-consistent charges using similar
+parameters to our DFT calculations. This included charge convergence criteria of 2.72 × 10−5 eV
+(10−6 au), a force convergence of 10−2 eV/ ˚A for all geometry optimizations, and 2×2×2 k-point
+mesh for all calculations.
+TABLE I: Ground state energies relative to diamond (∆Ediam) in eV/atom and nearest neighbor
+distances (NN) in ˚A for the Si polymorphs considered in this work.
+diamond
+bcc
+simple cubic
+graphene
+bc8
+NN ∆Ediam NN ∆Ediam NN ∆Ediam NN ∆Ediam NN ∆Ediam
+pbc/ChIMES
+2.37
+0.00
+2.67
+0.55
+2.53
+0.30
+2.23
+0.70
+2.37
+0.14
+siband/ChIMES 2.36
+0.00
+2.65
+0.53
+2.54
+0.31
+2.26
+0.59
+2.39
+0.15
+DFT
+2.37
+0.00
+2.68
+0.54
+2.53
+0.30
+2.25
+0.65
+2.39
+0.16
+In order to test the applicability of our ChIMES/DFTB models to different of Si phases, we have
+computed the relative energies and nearest neighbor distances for several polymorphs, including
+those in our training set as well as the bc8 phase (Table I). Our results indicate strong agreement
+with DFT for both models. We observe close agreement for all properties for both pbc/ChIMES
+and siband/ChIMES, where the energy of each phase relative to the diamond ground-state tends
+to agree with DFT within 0.01 eV, and the subsequent nearest neighbor distances agree within
+0.01 − 0.02 ˚A. The graphene phase is a small exception, where pbc/ChIMES yielded a relative
+energy of 0.70 eV/atom and siband/ChIMES a relative energy of 0.59 eV, compared to a value of
+0.65 eV for DFT. However, both models still yield the correct energetic ordering of the phases.
+Similar to previous efforts34,44, we have determined cold energy curves under isotropic com-
+pression and expansion for all phases in this study (Fig. 1).
+Overall, both pbc/ChIMES and
+siband/ChIMES yield close agreement with DFT. Both models have particularly close agreement
+for the diamond and simple cubic phases. The siband/ChIMES model exhibited a small oscil-
+lation in the bcc cold curve at a nearest neighbor distance of 2.7 ˚A which is not present in the
+pbc/ChIMES result. However, the agreement with DFT is reasonable for both models. The largest
+disagreement for pbc/ChIMES is with graphene, where it yields a more positive curvature at ex-
+panded densities, whereas siband/ChIMES yields closer agreement to DFT overall. Both models
+12
+
+−5.6
+−5.4
+−5.2
+−5
+−4.8
+−4.6
+−4.4
+−4.2
+−4
+ 2.2
+ 2.3
+ 2.4
+ 2.5
+ 2.6
+ 2.7
+ 2.8
+ 2.9
+ 3
+Energy/atom (eV)
+NN (Å)
+(a) pbc/ChIMES
+−5.6
+−5.4
+−5.2
+−5
+−4.8
+−4.6
+−4.4
+−4.2
+−4
+ 2.2
+ 2.3
+ 2.4
+ 2.5
+ 2.6
+ 2.7
+ 2.8
+ 2.9
+ 3
+Energy/atom (eV)
+NN (Å)
+(b) siband/ChIMES
+FIG. 1: Cold curves for several silicon polymorphs from pbc/ChIMES and siband/ChIMES
+DFTB models (points) compared to results from DFT (solid lines). The black curves correspond
+to the diamond phase, blue to bcc, red to simple cubic, and the green to graphene. The orange
+marks correspond to the bc8 phase and were not a part of the training set.
+predict very similar agreement for the bc8 phase, where each yielded a small oscillation in the
+cold curve around 2.5 ˚A. This is likely due to insufficient sampling of these Si-Si distances and
+bonding environments in our training set. Regardless, these results indicate strong agreement for
+energy vs. volume relationships, which could indicate accurate force prediction from each model.
+We now assess the force output from each model through comparison of the resulting vibra-
+tional density of states (VDOS) for the diamond phase to results from DFT (Fig. 2). These were
+computed from Fourier Transform of the velocity autocorrelation function which was determined
+from MD simulations at constant volume-temperature (NVT), conducted at 600 K, using a Nos´e-
+Hoover thermostatted chain80–82 and run for 15–20 ps using a timestep of 1 ps. Our results for
+pbc/ChIMES indicate fairly close agreement with DFT. Prediction of the lowest lying vibrational
+peak is off by only ∼7 cm−1, with a value of 134 cm−1 compared to a value of 127 cm−1 from
+DFT. DFT yields a small peak at 231 cm−1 which appears as a broad, higher intensity shoulder
+at 224 cm−1 in the pbc/ChIMES spectrum. The remaining peaks in the spectrum show similarly
+strong agreement with some variation in the intensity of the peaks, including accurate prediction
+from pbc/ChIMES of the vibron peak at 450 cm−1 compared to a frequency of 453 cm−1 from
+DFT.
+13
+
+−20
+ 0
+ 20
+ 40
+ 60
+ 80
+ 100
+ 120
+ 140
+ 160
+ 180
+ 200
+ 100
+ 200
+ 300
+ 400
+ 500
+Intensity
+Frequency (cm−1)
+FIG. 2: Vibrational density of states for the Si diamond phase, computed at 600 K. The red line
+corresponds to pbc/ChIMES. the blue line to siband/ChIMES, and the black dashed line to DFT.
+In contrast, siband/ChIMES shows slightly less accurate agreement with DFT overall. The
+agreement for the lowest vibrational peak is fairly close, with a frequency of 120 cm−1. The
+remainder of the siband/ChIMES spectrum yields an accurate overall shape of the VDOS, though
+with some errors in peak positions and intensities. There is some deviation in the siband/ChIMES
+spectrum for next two vibrational peaks, where we observe a frequency of 173 cm−1 for the second
+lowest frequency peak compared to a value of 188 cm−1 from DFT and a frequency of 217 cm−1
+for the low intensity peak after that compared to the previously mentioned DFT peak at 231 cm−1.
+The siband/ChIMES spectrum yields a close match in intensity and frequency with DFT for the
+VDOS peak at 344 cm−1. However, the subsequent two peaks are red shifted in frequency and
+lower in intensity, with values of peak positions of 413 and 472 cm−1, compared to values of 396
+and 453 cm−1 from DFT. The improved VDOS determination from pbc/ChIMES could be due
+in part to its parameterization with density superposition, which has been shown to yield more
+accurate predictions over potential superposition36. We note that these peak position differences
+discussed here correspond to small changes in energy, where 20 cm−1 corresponds to ∼ 2.5 ×
+10−3 eV. Hence, it is possible that siband/ChIMES will still yield sufficiently accurate forces for
+some applications.
+14
+
+(a) Vacancy
+(b) Tetrahedral
+(c) Hexagonal
+FIG. 3: Images of the diamond phase point defects investigated in this study. All defects are
+shown as a red sphere for the sake of clarity.
+TABLE II: Defect formation energies for the Si diamond phase. All energies are in eV.
+Defect
+pbc/ChIMES siband/ChIMES DFT (PBE)
+vacancy
+3.45
+4.60
+3.84
+tetrahedral
+5.11
+4.88
+3.84
+hexagonal
+5.87
+4.79
+3.61
+Finally, we have computed defect formation energies from our DFTB/ChIMES models (Fig. 3).
+Here, we have investigated a single Si atom vacancy as well as an interstitial atom in either a
+hexagonal or tetrahedral site, which were determined from use of the pymatgen software suite83.
+The tetrahedral interstitial site occurs where an additional Si atom is coordinated by four atoms
+from the lattice, whereas the hexagonal interstitial site occurs when the additional Si atom re-
+sides in a hexagonal opening within the lattice. The defect formation energy Eform is computed
+as Eform = Edef − NdefEdiam, where Edef is the total energy of the defect containing system,
+Ndef is the number of Si atoms in that configuration, and Ediam is the energy per atom of the
+perfect diamond phase. Similar to previous Si DFTB efforts44, calculations were initialized from
+an optimized 216 atom supercell where we retained a Monkhorst-Pack mesh of 2 × 2 × 2, after
+which we created the point defect and optimized the ionic positions of each configuration using
+the same k-point mesh. Our results indicate some agreement with previous PBE-DFT calculations
+from Ref. 44. The pbc/ChIMES model agrees with the DFT vacancy energy within 0.4 eV, but
+yields results that are 1–2 eV too high for both interstitial energies. In particular, the three defect
+energies from pbc/CHIMES differ over a range of over 2.4 eV, with the both interstitial energies
+15
+
+yielding larger results than that of the vacancy. In contrast, the result from DFT all lie relatively
+close together (within a range of 0.23 eV) and DFT exhibits equal formation energy values for
+the vacancy and tetrahedral interstitial. It is likely that the interstitial energies would be decreased
+with full accounting of d-orbital off-site interactions, which are absent in the original pbc-0-3
+parameter set. The siband/ChIMES model yields defect formation energies that are consistently
+∼1 eV too high relative to DFT. However, the siband/ChIMES results differ over an energy range
+of 0.28 eV, yielding improved agreement with DFT in this respect. It is likely that there would
+be some variation in DFT results depending on the choice of exchange-correlation function and
+possible inclusion of a dispersion energy correction.
+Overall, our we able to create two new DFTB/ChIMES models that more closely approach a
+single-purpose approach for silicon phases under different conditions. The pbc/ChIMES model ap-
+pears to yield a somewhat improved description of atomic forces, whereas as the siband/ChIMES
+model yields more systematically consistent defect formation energies that could make it prefer-
+able for some calculations. As mentioned, some of the limitations of the pbc/ChIMES model could
+possibly be overcome through inclusion of d-orbital two-center interactions in the corresponding
+Slater-Koster file. Regardless, we now provide a repulsive energy for the siband-1-1 parameter
+set, which will allow its use for structural relaxations and/or dynamics calculations in addition to
+its accuracy for electronic properties. It is possible that the slightly longer cutoff radius for our
+ChIMES ERep could be mitigated through optimization of the choice of DFTB confining radii
+(discussed in the next section). Future improvement of these models could also involve inclusion
+of data from MD simulations of amorphous or defect containing systems at different temperatures
+and pressures.
+B.
+Semi-automated Workflow for DFTB/ChIMES Model Creation
+In this subsection, we summarize previous work on TiH236 which indicates the utility in using a
+ChIMES ERep in a semi-automated fashion to screen for optimal confining radii in a Slater-Koster
+file parameterization. TiH2 has a number of industrial uses as a functional material, including
+in hydrogen storage alloys84, superconductors85, and as a blending agent for porous foams86. Its
+ground-state structure exhibits face-centered-cubic (fcc) symmetry, with the (111) facet computed
+to have the lowest surface energy (Fig. 4). Several adsorption sites are illustrated, including Top
+(directly above a Ti atom), Hollow (in an interstitial cavity), and several Bridge sites (existing in
+16
+
+between Ti-Ti and H-H nearest neighbors) sites. TiH2 is a somewhat ideal system for DFTB model
+development in that DFT calculations on small supercells are relatively tractable, which allows for
+straightforward validation. DFT calculations though are generally too computationally inefficient
+for the larger supercells needed to model grain boundaries and crystalline defects at sufficiently
+low concentration, allowing for several applications of a new TiH2 DFTB model in future studies.
+Ti
+H
+x
+o
+o
+o
+o
+o
+o
+o
+x2
+x1
+(111) surface
+(011) surface
+Bulk
+FIG. 4: Pictures of TiH2 bulk and surfaces. The left panel shows the bulk fcc lattice. The middle
+panel shows the (111) crystalline surface the Top (marked with an ‘O’) and Hollow (‘X’)
+adsorption sites indicated. The right panel shows the (011) crystalline surface with the Top (‘O’),
+Bridge-1 (‘X1’) and Bridge-2 (‘X2’) sites indicated. Reprinted with permission from Journal of
+Chemical Theory and Computation 2021 17 (7), 4435-4448. Copyright 2021, American
+Chemical Society.
+Here, we have leveraged rapid ChIMES ERep optimzation by creating a workflow that allowed
+us to perform a semi-exhaustive search of all DFTB and ChIMES hyperparameters (Fig. 5). We
+first compute a matrix of thirty Slater-Koster files from titanium wavefunction confining radii
+(RTi
+ψ ) and density confining radii (RTi
+n ) sampled over a range of 3.2 ≤ RTi
+ψ
+≤ 5.0 au and
+6.0 ≤ RTi
+n
+≤ 17.0 au. Hydrogen interaction parameters were taken from the miomod-hh-0-1
+parameter set. Model down selection could then be performed over the entire grid Slater-Koster
+tables through comparison to our selected validation data, which allowed us to determine optimal
+ChIMES polynomial orders and the LASSO regularization parameter.
+For this work, our DFT training set consisted of molecular dynamics simulations of unit cell
+configurations (12 atoms total), run for 5 ps at 400 K with simulation cells initially optimized to
+pressures in a range from −8 to 100 GPa. All MD calculations were run in the constant temperature
+and volume (NV T) ensemble with Nos´e-Hoover thermostat chains80–82 and a timestep of 0.2 fs.
+The slightly elevated temperature and wide pressure range including negative pressure were chosen
+in order to yield a broad sampling of the underlying potential energy surface. Atomic forces and
+the diagonal of the stress tensor were then sampled from MD configurations at fixed time intervals
+17
+
+bbSelect !", $!, $"; 
+Create SKF files.
+Compute DFTB training set: 
+⃗&DFT − ⃗&DFTB (no ERep)
+(##,DFT −(##,DFTB (no ERep)
+Desired accuracy 
+achieved?
+Validation set:
+•
+Bulk: lattice const., VDOS, 1H and 2H 
+vacancies.
+•
+(001), (011), and (111) surface energies
+•
+Eads on (011) and (111) surfaces (5 total)
+•
+(011) and (111) surface and subsurface 
+H vacancy energies (8 total)
+Choose ChIMES 2B, 3B, 
+4B orders, cutoff radii 
+and determine )$%&.
+Yes
+No
+Complete
+Compute 
+DFT-MD 
+data
+FIG. 5: Flowchart for creation of DFTB Erep models through ChIMES force field
+parameterization. Reprinted with permission from Journal of Chemical Theory and Computation
+2021 17 (7), 4435-4448. Copyright 2021, American Chemical Society.
+of ∼ 160 fs in order ensure configurations were as statistically uncorrelated as possible. This
+yielded up to 30 MD snapshots for each pressure. Inclusion of system energies in our training data
+did not appear to improve the quality of our optimization and hence were omitted. In addition, in
+order to sample hyper- and hypo-coordinated configurations in the system, we included MD data
+for a unit cell with a single hydrogen interstitial or single vacancy site, each run for 5 ps. This
+yielded a total of 153 unit cell-sized configurations for our training set. Validation calculations
+for all of our DFTB/ChIMES models were performed on the bulk lattice constant, single and
+double hydrogen vacancy energies, and the vibrational density of states. We also validated our
+models against a number of surface properties, including the surface energies of the (001), (011)
+and (111) facets, five different hydrogen adsorption energies on the (011) and (111) surfaces, and
+surface and subsurface hydrogen vacancy energies on the same two facets. Validation data for
+hydrogen interactions with the (001) surface were omitted from our study due to the presence of a
+significant surface dipole on this facet.
+Once again, all DFT calculations were performed with VASP using PAW pseudopotentials and
+PBE. We found our results to be converged with a planewave cutoff of 400 eV and an energy
+18
+
+convergence criteria of 10−6 eV, both of which were used for the results reported here. Fourth
+order Methfessel-Paxton smearing87 was used with a value of 0.13 eV for all geometry and cell
+lattice optimizations in order to ensure energy convergence without dependence on the electronic
+smearing temperature. The Mermin functional78 with the same electronic temperature was used for
+all MD calculations in order to avoid spurious forces due to possible negative occupation numbers
+from the Methfessel-Paxton approach. Brillouin Zone sampling for all TiH2 unit cell calculations
+was performed with a 10 × 10 × 10 k-point mesh, whereas we used a mesh of 5 × 5 × 5 for
+32 formula unit (96 atom) bulk calculations. We used system sizes of 168 atoms/7 layers for
+the (001) surface, 144 atoms/6 layers for the (011) surface, and 192 atoms/8 layers for the (111)
+surface, each with a vacuum of 20 ˚A and a k-point mesh of 5×5×1 in the direction of the surface.
+DFTB+ calculations were performed using self-consistent charges (SCC)22 and charge conver-
+gence criteria of 2.72 × 10−5 eV (10−6 au). Inclusion of an external van der Waals correction31,88
+is beyond the scope of our present study. We have performed “shell-resolved” SCC calculations,
+where separate Hubbard U parameters were determined for each orbital angular momentum shell.
+All minimum and cutoff radii for the ChIMES ERep were set to include the first coordination shell
+sampled in our training set, only: 2.5 ≤ rTiTi ≤ 3.5 ˚A and 1.5 ≤ rHTi ≤ 2.5 ˚A . We use values
+of λTiTi = 3.0 ˚A and λHTi = 2.0 ˚A for the Morse-like coordinate transforms. H-H repulsive
+interaction were not sampled in our training set and were thus also taken from the miomod-hh-0-1
+parameter set.
+Our results for a subset of our validation data (Fig. 6) allow us to describe general trends
+regarding the confining radii. We observe an approximate linear relationship between RTi
+ψ and
+RTi
+n in terms of the accuracy of the E111 energy, where the most accurate surface energy results
+from either small or large choice for both radii. All of the DFTB/ChIMES models created in this
+iteration tend to under-predict the (E001/E111) ratio (i.e., the ratio of highest to lowest surface
+energies in our study) relative to our DFT calculations, where we observe values of 1.35–1.44
+compared to the DFT ratio of 1.70. We note that is is likely in part due to the surface dipole
+moment present in our construction of the (001) facet. In addition, our results indicate a much
+smaller dependence on choice of RTi
+n for a given RTi
+ψ . We note that there can be strong dependence
+of the surface energies on choice of DFT functional (e.g., Ref. 89), although the relative energetic
+ordering tends to be consistent.
+Our final set of hyper-parameter values includes {RTi
+ψ = 3.6 au, RTi
+n = 6.0 au} and {O2B = 8,
+O3B = 4}, optimized with LASSO/LARS and regularization of α = 10−3. This model yields
+19
+
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+ 18
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+RTi
+n (au)
+RTi
+ψ (au)
+−0.2
+−0.15
+−0.1
+−0.05
+ 0
+ 0.05
+ 0.1
+ 0.15
+Fractional Deviation of E111
+ 6
+ 8
+ 10
+ 12
+ 14
+ 16
+ 18
+ 3
+ 3.5
+ 4
+ 4.5
+ 5
+RTi
+n (au)
+RTi
+ψ (au)
+−0.35
+−0.34
+−0.33
+−0.32
+−0.31
+−0.3
+−0.29
+−0.28
+−0.27
+−0.26
+∆(E001/E111)
+FIG. 6: Results for sweep of values of RTi
+ψ and RTi
+n , where the ChIMES ERep was determined
+with a 2B order of 12 and 3B order of 8. The top panel corresponds to the fractional deviation of
+the surface energy,
+�
+EDFTB
+111
+− EDFT
+111
+�
+/EDFT
+111 , and the middle panel to the deviation of
+(E001/E111) relative to DFT. Reprinted with permission from Journal of Chemical Theory and
+Computation 2021 17 (7), 4435-4448. Copyright 2021, American Chemical Society.
+RMS errors of 0.076 eV/ ˚A for hydrogen forces, 0.056 eV/ ˚A for titanium forces, and 0.35 GPa for
+the stress tensor diagonal. Results for bulk properties indicate that DFTB/ChIMES yields a lattice
+constant with errors of only ∼0.4% and 1.0% from DFT and experiment90, respectively. However,
+our model yields a hydrogen bulk vacancy energy (Evac) that is ∼0.5 eV too small. We found
+that a systematic ∼0.5 eV underestimation of vacancy energies in a variety of environments and
+concentrations was typical for all ChIMES parameterizations created in this work, which could be
+rectified with improved training data or adaptations to DFTB such as the inclusion of multi-center
+terms in the Hamiltonian.91.
+Overall, our final model yields accurate surface energies for all three low-index facets investi-
+gated in this study (Table III). In particular, the E011 and E111 values are nearly identical to those
+from DFT. The E001 value from DFTB/ChIMES is around 17% lower than than that for our DFT
+calculations (0.114 vs. 0.136 eV/ ˚A2). This could be due in part to the internal electric field on the
+(001) surface configuration studied here, as mentioned. DFTB generally can underestimate surface
+electrostatic interactions due to its determination of atom-centered point charges only in Coulom-
+bic interactions92. Our DFTB/ChIMES results show similarly strong agreement with hydrogen
+surface adsorption energies (Table IV). We compute the correct energetic ordering of adsorption
+on the (111) Top and Hollow sites, though the Hollow site energy is 0.35 eV smaller than that from
+20
+
+DFT. We see similar agreement with DFT for the (011) surface. Here, DFTB/ChIMES show close
+agreement for Top site adsorption with a difference of only 0.05 eV from DFT. Our model yields
+Bridge-1 and Bridge-2 adsorption energies that differ from DFT by 0.29 eV and 0.21 eV, respec-
+tively, and incorrectly predicts that the Top site is the lowest energetically of the three. Regardless,
+these values are similar in energy for all surface sites and we have overall favorable agreement.
+TABLE III: TiH2 surface energies (in eV/ ˚A2). Reprinted with permission from Journal of
+Chemical Theory and Computation 2021 17 (7), 4435-4448. Copyright 2021, American
+Chemical Society.
+Surface DFTB/ChIMES DFT
+111
+0.080
+0.080
+011
+0.105
+0.101
+001
+0.114
+0.136
+TABLE IV: Surface hydrogen adsorption energies on TiH2 surface sites (in eV). Reprinted with
+permission from Journal of Chemical Theory and Computation 2021 17 (7), 4435-4448.
+Copyright 2021, American Chemical Society.
+Surface
+Site
+DFTB/ChIMES DFT
+111
+Top
+-1.888
+-1.760
+Hollow
+-2.081
+-2.440
+011
+Top
+-2.383
+-2.332
+Bridge-1
+-2.154
+-2.442
+Bridge-2
+-2.132
+-2.342
+Our results indicate DFTB/ChIMES models can be accurately determined based on relatively
+small training data (unit cell MD calculations in this work), even for physically complex sys-
+tems such as those containing surface chemistry. Further refinement of our TiH2 model could
+involve inclusion of training data from additional phases and thermodynamic state points. Re-
+gardless, our current effort yields accurate results for bulk and surface TiH2 properties, and our
+model shows strong transferability to bulk α-Ti and gas phase TiH4 (not shown here). The small
+training set could yield significant advantages for computationally challenging systems such as
+21
+
+magnetic materials and their interfaces, where DFT data is limited and difficult to generate. Over-
+all, our DFTB/ChIMES approach can have particular impact on myriad of research areas, such as
+interpretation of imaging and spectroscopy studies on bulk and interfacial systems, where there is
+traditionally a strong coupling with atomistic simulation approaches.
+C.
+∆-learning to Enhance the Accuracy of DFTB for Organic Materials
+In this subsection we review our recent efforts to leverage a high-level quantum chemical
+database to create an “out-of-the-box” model with accuracy beyond standard DFT approaches
+(e.g., PBE) that is generally applicable to many organic molecular systems56. In this work, we have
+used the ANI-1x quantum chemical data set93,94 to create a DFTB/ChIMES model that approaches
+hybrid-functional and/or coupled cluster accuracy. Here, ChIMES is used as a ∆-learning po-
+tential where we have included it as an extra energy term to the 3ob-3-1 parameterization40,95,
+which includes third-order charge fluctuation terms in the DFTB energy. This parameterization is
+known to yield reliable accuracy for many organic molecules and thus was a reasonable starting
+point for our efforts. We have found that the advantage of ChIMES over a neural network ap-
+proach is two-fold: (1) the training set requirements of ChIMES is significantly lower, where only
+a small fraction of the ANI-1x dataset was required to achieve a high degree of accuracy, and (2)
+our ChIMES potential required two-order of magnitude fewer parameters than several recent NN-
+based semi-empirical approaches. These effects allow for a much easier to parameterize model
+that is less likely to be hampered by overfitting.
+The original ANI-1x database was developed for the creation of ML-based general-purpose
+organic potentials where the data set was determined through an active learning process94, result-
+ing in approximately 5 million molecular equilibrium and non-equilibrium configurations. Our
+∆-learning optimization used an iterative approach by first creating a subset of ANI-1x called
+“sub ANI-1x” that only contained results computed from CCSD(T) (coupled-cluster consider-
+ing single, double, and perturbative triple excitations) and using a well-known hybrid functional,
+wB97X96. This corresponded to 459,464 molecular confirmations from computed from 1895
+unique molecules, or ∼10% of the original ANI-1x database. We note that there are no atomic
+force data from CCSD(T)/CBS calculations. Hence, we used wB97X results computed with a
+large basis set (def2-TZVPP) data for fitting purposes, with the remainder of the data set available
+for validation.
+22
+
+We then used an iterative approach to ChIMES optimization (Fig. 7) where we first randomly
+selected only 1% of sub ANI-1x and performed an initial ChIMES optimization. Validation cal-
+culations agains the remainder of sub ANI-1x resulted in some large deviations in the computed
+energies and forces. We then selected an additional equivalent of 1% of the data set from con-
+figurations with the highest force deviations and added them to our training set and repeated the
+process, where each increment of the training process would include the equivalent of an additional
+1% of sub ANI-1x. Our DFTB/ChIMES ∆-learning was converged after three iterations of our
+optimization scheme, using only 3% of sub ANI-1x or 0.3% of the original ANI-1x database. Our
+model was ultimately validated against the entire sub ANI-1x data set, though its size is somewhat
+arbitrary and it is possible that a smaller subset of ANI-1x could have been used for our purposes.
+-2000 -1000
+0
+1000
+2000
+FDFT(kcal/mol/Å)
+-2000
+-1000
+0
+1000
+2000
+FDFTB/ChIMES(kcal/mol/Å)
+-2000 -1000
+0
+1000
+2000
+FDFT(kcal/mol/Å)
+-2000
+-1000
+0
+1000
+2000
+-2000 -1000
+0
+1000
+2000
+FDFT(kcal/mol/Å)
+-2000
+-1000
+0
+1000
+2000
+-20
+0
+20
+40
+60
+EDFT(kcal/mol/atom)
+-20
+0
+20
+40
+60
+EDFTB/ChIMES(kcal/mol/atom)
+circle 0
+1% train and 99% validation
+-20
+0
+20
+40
+60
+EDFT(kcal/mol/atom)
+-20
+0
+20
+40
+60
+circle 1
+2% train and 98% validation
+-20
+0
+20
+40
+60
+EDFT(kcal/mol/atom)
+-20
+0
+20
+40
+60
+circle 2
+3% train and 97% validation
+a)
+b)
+c)
+d)
+e)
+f)
+FIG. 7: Comparison of energies per atom (top panels) and forces (bottom panels) predicted by
+DFT (wB97X) and DFTB/ChIMES for all configurations in the validation set. The dataset used
+here is ‘sub ANI-1x’, ∼10% of the full ANI-1x. Reprinted with permission from J. Phys. Chem.
+Lett. 2022 13 (13), 2934-2942. Copyright 2022, American Chemical Society.
+Our final model used ChIMES polynomial orders of {2B = 24, 3B = 10, 4B = 0} with a
+somewhat long radial cutoff of 4.0 ˚A used for all atom pairs. This longer cutoff helped account
+for some dispersion interactions that would otherwise be absent from standard DFTB calculations,
+though future efforts will involve shorter cutoffs combined with a dispersion interaction model.
+23
+
+Further details about our ChIMES model for organics can be found in the Supporting Information
+in Ref. 56. Ultimately, our DFTB/ChIMES model resulted in 5546 parameters and was trained to
+∼372k data points. This is in contrast to the recently developed AIQM1 semi-empirical quantum
+model, which utilizes an NN trained to the entire ANI-1x data set, resulting in 322,660 parameters.
+Similarly, a recent DFTB-NN approach using deep-tensor neural networks used a training set of
+∼800k data points, resulting in 228,865 parameters.
+TABLE V: Performance of DFTB and DFTB/ChIMES in predicting reference energies and/or
+atomic forces in the GDB-10to13, ISO34, and GDML data set. The MAE and RMSE for the
+energies and forces (labeled with subscripts ‘E’ and ‘F’) are in kcal/mol and kcal/mol- ˚A,
+respectively. Reference molecular energies and atomic forces in the GDB-10to13 data set are at
+the wB97X/6-31G* level of theory. Isomerization energies in the ISO34 data set are a mixture of
+experimental- and CCSD(T) extrapolation energies. The CCSD(T)/cc-pVTZ atomic forces of
+2000 configurations of ethanol in the GDML data set are used for comparison. Reprinted with
+permission from J. Phys. Chem. Lett. 2022 13 (13), 2934-2942. Copyright 2022, American
+Chemical Society.
+GDB-10to13
+ISO34
+GDML
+method
+MAEE/RMSEE MAEF/RMSEF MAEE/RMSEE MAEF/RMSEF
+DFTB
+9.10/11.70
+6.34/9.85
+3.69/4.96
+4.52/6.12
+DFTB/ChIMES
+3.57/4.72
+3.62/5.33
+2.06/2.56
+2.72/3.61
+ANI-197
+3.12/4.74
+3.96/7.09
+-
+-
+ANI-1x97
+2.30/3.21
+3.67/6.01
+-
+-
+DFTB-NNrep98
+-
+-
+2.21/3.30
+-
+PBE098
+-
+-
+1.82/2.48
+-
+We then tested the transferability of our DFTB/ChIMES model through comparison to different
+quantum chemical data that were computed at the wB97X or CCSD(T) level but were not a part
+of ANI-1x (Table V). For example, the GDB-10to13 data set97 consists of the molecular energies
+and forces at the wB97X level of nearly 3000 molecules containing 10-13 C, N, or O atoms for a
+total of 47,670 configurations. Our DFTB/ChIMES model exhibits a 60% reduction in the mean
+average error (MAE) and RMSE error in the energies and a 45 % decrease in the forces over
+24
+
+standard DFTB. The accuracy of DFTB/ChIMES is similar to values from the ANI-1 and ANI-
+1x neural network interatomic potentials97 (i.e., stand-alone potentials without explicit quantum
+mechanical elements), and are smaller than the variations between wB97X itself and higher levels
+of theory such as CCSD(T) and MP2 (4.9/5.9 kcal/mol for energies and 4.6/5.9 kcal/mol- ˚A for
+forces)93.
+Our DFTB/ChIMES model is validated against additional CCSD(T) reference data from the
+ISO34 data set99, which consists of energies of 34 isomers containing the elements C, H, N,
+and O. We observe that the accuracy of DFTB/ChIMES is much better than that for standard
+DFTB, is slightly improved over that from DFTB-NNrep, and approaches the PBE0 data given
+in Ref. 98. To test the performance of our model on high accuracy force data specifically, we
+compare DFTB/ChIMES with the CCSD(T)/cc-pVTZ data for 2000 configurations of ethanol in
+the GDML data set100 (54,000 data points total). Again our DFTB/ChIMES gives an improvement
+over standard DFTB as MAE and RMSE are both reduced by ∼40%. A direct force comparison to
+DFTB-NNrep or the ISO34 reference was unavailable. Additional validation of our model included
+calculation of the n-butane dihedral potential and correct prediction of the energetic ordering of
+coumarin molecular crystals.
+We have also validated DFTB/ChIMES against vibrational frequencies of 342 gas-phase
+molecules from the Computational Chemistry Comparison and Benchmark Database or CC-
+CBDB (https://cccbdb.nist.gov/), computed with MP2/cc-pVTZ and wB97XD (with dispersion
+correction), amongst other methods (Fig. 8). Here, DFTB/ChIMES yields errors in the frequency
+prediction of MAE/RMSE = 36/61 cm−1, indicating improved accuracy over PBE and with similar
+accuracy to accuracy to wB97XD. In all of our validation tests, DFTB/ChIMES shows marked
+improvement over standard DFTB and PBE, and shows similar accuracy to results from wB97X or
+other higher-levels of theory. Further details of all validation calculations are provided in Ref. 56.
+Lastly, though the DFTB/ChIMES model developed here is trained on gas phase molecular
+data, we have also tested its performance in reproducing the structural properties of bulk graphite
+and diamond. We compare predicted density and lattice parameters from different methods in
+Table VI.
+For graphite, all computational models considered here give an accurate descrip-
+tion of the in-plane lattice parameters. DFTB and PBE overestimate the interlayer separation
+(c/2) by over 25% and 30%, respectively, due to their under-prediction of dispersion interactions.
+DFTB/ChIMES predicts the lattice parameters and density in excellent agreement with the exper-
+imental value, with a deviation of less than 1%. For diamond, the computed values using DFTB,
+25
+
+0
+1000
+2000
+3000
+4000
+Frequency (cm
+-1)
+Distribution
+MP2
+ωB97XD
+DFTB/ChIMES
+PBE
+DFTB
+FIG. 8: The distribution of the calculated frequency values using DFTB and DFTB/ChIMES for
+342 neutral molecules taken from the CCCBDB database. The MP2 and DFT (PBE and
+wB97XD) calculations using cc-pVTZ basis set in the CCCBDB are selected for comparison.
+Reprinted with permission from J. Phys. Chem. Lett. 2022 13 (13), 2934-2942. Copyright 2022,
+American Chemical Society.
+DFTB/ChIMES, and PBE-DFT differ by ∼1% from experimental values for lattice parameters
+and ∼3% for the density.
+Ultimately, we have shown that ChIMES can be used to extend DFTB to hybrid functional
+accuracy or greater. DFTB/ChIMES has the capability of reproducing vast quantities of high-level
+reference data while requiring only a small fraction of it for training. On the basis of the results
+presented here, DFTB/ChIMES represents a promising direction for developing general purpose
+quantum models that are applicable to a wide range of materials and conditions. The small training
+set required as well as the small number of potential parameters relative to neural network methods
+could yield significant advantages for future development of computational efficient models with
+up to coupled cluster accuracy. The ease of parameterization and transferability of DFTB/ChIMES
+26
+
+TABLE VI: Comparison of predicted density and lattice parameters of graphite and diamond for
+DFTB, DFTB/ChIMES, PBE-DFT with experimental data. Reprinted with permission from J.
+Phys. Chem. Lett. 2022 13 (13), 2934-2942. Copyright 2022, American Chemical Society.
+phase
+method
+density (g/cm3) a( ˚A) c/2( ˚A)
+graphite Expt.101
+2.26
+2.462 3.356
+PBE-DFT102
+1.71
+2.470 4.420
+DFTB/ChIMES
+2.25
+2.461 3.379
+DFTB
+1.77
+2.474 4.248
+diamond Expt.101
+3.51
+3.567
+PBE-DFT70
+3.48
+3.580
+DFTB/ChIMES
+3.42
+3.600
+DFTB
+3.42
+3.600
+allows for high-level quantum theory accuracy in systems where traditional wavefunction or hybrid
+functional methods are far too computationally intensive for intensive use.
+IV.
+DISCUSSION AND FUTURE WORK
+ChIMES was initially developed as a method for creating many-body force fields for molecular
+dynamics simulations. However, it has also proven robust as a repulsive energy for DFTB models,
+where the standard two-center approach for both quantum mechanical and repulsive terms can be
+insufficient for many systems. The strength in ChIMES as an element of a semi-empirical quantum
+model or MD model lies in its use of linear combinations of many-body Chebyshev polynomials,
+where the nearly optimal nature of the polynomials as well as the linear least-squares fitting allow
+for rapid optimizations that require far fewer parameters and significantly smaller data sets than the
+neural network models reviewed here. In addition, ChIMES adds very little extra computational
+time to DFTB calculations, where the matrix diagonalization and SCC convergence use the vast
+majority of the CPU effort.
+Future work will involve extending ChIMES to systems with four or more elements, where de-
+velopment of training sets and proper validation approaches remains an open question. It is likely
+that these ChIMES models will require larger data sets and the potentials themselves will have
+27
+
+significantly more parameters than those presented in this work due to the combinatorial effect
+of forming many-body clusters with different possible combinations of elements. Determination
+of ERep for these systems will likely yield significant advantages over pure interatomic potentials
+due to the short-ranged nature of the repulsive energy as well as the general accuracy of the quan-
+tum mechanical parts of DFTB. Both of these considerations make creation of DFTB/ChIMES
+model in general more tractable than optimizing ChIMES on its own as an atomistic force field.
+DFTB/ChIMES can serve as either a stand-alone model for running dynamics and determining
+physical and chemical properties of a system, or as a surrogate for DFT in a “boot-strapping” op-
+timization, where it can serve to generate reasonably high fidelity training data for pure ChIMES
+MD models. Overall, our approach can be used to enhance the speed of quantum accurate pre-
+dictions for both molecular and condensed matter systems, where there is a historic reliance on
+computationally intensive quantum simulations for predictions of chemical and physical properties
+related to experiments.
+ACKNOWLEDGMENTS
+This work performed under the auspices of the U.S. Department of Energy by Lawrence Liv-
+ermore National Laboratory under Contract DE-AC52-07NA27344. The assigned release number
+is LLNL-JRNL-XXXXXX.
+28
+
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