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+On using the generalized Langevin equation
+to model substrate phonons and their role in
+surface adsorption and desorption
+Ardavan Farahvash,† Mayank Agarwal,‡ Andrew Peterson,‡ and Adam P.
+Willard∗,†
+†Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts
+02139, USA
+‡Department of Chemical Engineering, Brown University, Providence, Rhode Island 02912, USA
+E-mail: [email protected]
+Abstract
+Surface vibrations are an important aspect in many gas-surface reactions and thus under-
+standing the action of these vibrations using a simple theoretical model is highly desirable.
+The generalized Langevin equation is one such model, as it can reduce all the aspects how vi-
+brations modulate the motion of a surface site into a single quantity: the memory kernel. Here
+we build on work originally done by Tully in developing the generalized Langevin oscillator
+(GLO) model of surface vibrations by calculating the memory kernel directly from atomistic
+models. We show that the memory kernel has a universal bimodal form due to coupling to
+both low-energy acoustic modes and modes near the Debye frequency. We study the size de-
+pendence of these modes and argue that the acoustic modes are frozen in limit of macroscopic
+lattices. We then use this insight to illustrate how finite size effects from nanoscale systems,
+such as those studied with atomistic simulations, can alter surface adsorption and desorption.
+1
+arXiv:2301.04873v1  [physics.chem-ph]  12 Jan 2023
+
+1
+Introduction
+Heterogeneous catalysis is the backbone of much of the modern chemical industry. Owing to this
+enormous utility, understanding the nature of molecular dynamics at catalyst interfaces and how
+molecular properties impact catalytic reactivity remains an important and outstanding scientific
+challenge. While the electronic degrees of freedom of both the reagent and substrate are central
+to understanding the binding free-energy, and thus reactivity per Sabatier’s principle,1,2 recent
+experiments and theoretical studies have highlighted the often underappreciated role of substrate
+vibrations in modulating the reaction dynamics at metal interfaces.3–12 For example, several stud-
+ies have shown that laser pulses can be used to indirectly excite surface phonons and subsequently
+increase desorption rates and reactivity.3–6,13 It has been shown that applying acoustic waves to
+a catalyst surface at an appropriate polarization and resonant frequency, can also greatly increase
+reactivity.7,12,14 It is not clear whether such enhancements of reactivity are due to an effective
+reduction of the binding energy, or due to kinetic factors which go beyond Sabatier’s purely ther-
+modynamic principle. In light of such experiments, an improved understanding of the ways in
+which substrate vibrations may modulate reactivity is greatly desired. In this paper, we employ
+the generalized Langevin equation to study substrate vibrations, focusing on how lattice phonons
+affect the motion of surface sites, and on the relationship between phonon-induced memory and
+surface sticking probabilities and desorption rates.
+The generalized Langevin equation, shown below, is an incredibly useful tool for understanding
+the motion of a single atom, or small collection of atom, within a lattice,
+˙p(t) = −dW
+dq (t)−
+� t
+0 K(t −τ)p(τ)dτ +R(t).
+(1)
+Here q and p are the position and momenta of the site of interest. W is the potential of mean
+force, the thermodynamic free-energy surface that results from coarse-graining the other lattice
+degrees of freedom. K(t) is the memory kernel, a time-dependent analog of the Markovian fric-
+tion constant, and R(t) is a random, Gaussian process whose autocorrelation function is related
+2
+
+to K by the second fluctuation dissipation theorem (FDT) K(t) = ⟨R(t)R(0)⟩
+mkBT
+. Adelman and Doll15
+were the first to discuss how the GLE could be used to model the effect of substrate phonons on
+a site within a purely harmonic lattice. Later, Tully developed the generalized Langevin oscillator
+(GLO) method, wherein the motion of surface site is described via a GLE with a memory kernel
+that is given by a single exponentially damped sinusoid.16 As we will discuss in greater detail in
+Section 2.3, such a memory kernel is equivalent to coupling the surface atom to a single dissipative
+(ghost) oscillator. Tully’s GLO method has seen much success as a computationally efficient way
+of modeling substrate dynamics, particularly in application to molecular beam scattering experi-
+ments.17–21 However, the choice of coupling to a single mode is arbitrary and limiting; in principle
+surface atoms should couple to each normal mode of the lattice.
+In this study, we extend Tully’s GLO model to allow for a memory kernel of arbitrary shape,
+and examine when and how the properties of the memory kernel affect adsorption and desorption.
+We call this model the lattice generalized Langevin equation (LGLE). In order to parameterize the
+memory kernel we use data taken from atomistic simulations. Crucially, in Section 4 we show
+that the qualitative properties of the memory kernel are independent of the details of the atomistic
+model used.
+The remainder of the paper will be organized as follows. In Section 2 we review the formal the-
+ory behind the LGLE, focusing on methods to parameterize the equation and briefly reviewing the
+extended variable transformation used to map the non-Markovian dynamics to a bath of dissipative
+harmonic oscillators. In Section 3 we will detail simulation methods used to generate data for this
+paper. In Section 4 we analyze memory kernels taken from atomistic simulations using different
+forcefields, metals, and solvation states of the lattice. Notably, we show that the memory kernel
+has a bimodal form arising from strong coupling to both coherent acoustic oscillations as well as
+modes near the Debye frequency. Finally, in Section 5 we discuss how the properties of the mem-
+ory kernel we calculated in the previous sections affect adsorption and desorption, highlighting
+systematic errors that can occur when using nanoscale simulations as a substitute for macroscopic
+systems.
+3
+
+2
+Theoretical Background
+While there are many approaches to parametrizing Eq.1 from atomistic data, we will discuss two
+in particular, henceforth termed the projection operator (PO) method and the correlation function
+(CF) method. The PO method is the approach employed in the formative works of Adelman
+and Doll and Tully, and involves little more than matrix operations involving the mass-weighted
+Hessian of the lattice. The CF method is commonly used in applications of the GLE to amorphous
+systems or liquid solutions.22–26 Both approaches are useful and will be employed in subsequent
+sections.
+2.1
+Projection operator method
+Our starting point in this method is to expand the potential energy surface, U(q1 ...qN), of the
+lattice to second order, such that the Hamiltonian may be written as,
+H = ∑
+i
+p2
+i
+2m + 1
+2∑
+ij
+∂ 2U
+∂qi∂qj
+(qi − ¯qi)(q j − ¯q j),
+(2)
+where qi are the positions of each lattice site, pi are the momentum, and ¯qi are the equilibrium
+positions. By introducing mass-weighted coordinates xi = √miqi and the mass-weighted Hessian
+D2
+i j =
+1
+√mim j
+∂ 2U
+∂qi∂qj the equation of motion may be written simply as,
+¨x = −D2x.
+(3)
+Two projection operators, P and Q = 1−P are then used to separate this equation in to a system
+(surface site) and bath (remaining lattice) subspace respectively,
+¨xP = −D2
+PPxP −D2
+PQxQ,
+(4)
+¨xQ = −D2
+QPxP −D2
+QQxQ,
+(5)
+4
+
+where xP = Px are the system degrees of freedom, xQ = Qx are the bath degrees of freedom and
+D2
+PP = PD2P, etc. In principle, these projection operators can take any form, so long as they
+obey the properties of idempotency P2 = P and orthogonality PQ = 0. However, if we wish xP
+to correspond to the displacement of a surface atom, it is most natural to choose P to be a matrix
+with ones on the diagonal for the indices corresponding to the coordinate(s) of interest and zeros
+elsewhere,
+P =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+1
+0
+0
+0
+�
+�
+�
+�
+�
+�
+�
+�
+�
+D2 =
+�
+�
+�
+�
+�
+�
+�
+�
+�
+D2
+PP
+D2
+PQ
+D2
+QP
+D2
+QQ
+�
+�
+�
+�
+�
+�
+�
+�
+�
+(6)
+For simplicity, let us allow xP to be a scalar xP corresponding to the displacement of a single
+surface atom along a single coordinate. One then proceeds by solving Eq.5 in terms of xP, substi-
+tuting that solution into Eq.4, and subsequently transforming back from mass-weighted to standard
+coordinates. The details of such steps may be found in several other papers. The final solution is
+of the form of Eq.1 with,
+K(t) = D2
+PQ
+cos(DQQt)
+D2
+QQ
+D2
+QP,
+(7)
+W(q) = m
+�
+−D2
+PP +K(0)
+�
+q2,
+(8)
+R(t)
+√m = D2
+PQ
+�
+cos(DQQt)xQ(0)+ sin(DQQt)
+DQQ
+˙xQ(0)− cos(DQQt)
+D2
+QQ
+D2
+QPxP(0)
+�
+.
+(9)
+Disregarding some mathematical subtleties with respect to the third term in Eq.9,27 K(t) and R(t)
+above do satisfy the FDT. Note that because K(t) determines the properties of R(t) via the FDT and
+also the determines the deviation of W from a fixed lattice, K(t) is arguably the most fundamental
+quantity in Eq.1, containing all the relevant information for how the bath modulates the system’s
+dynamics.
+Throughout the rest of this paper, we will often find it useful to analyze the Fourier transform
+of the memory kernel, K(ω), which is equivalent to the power spectrum of the noise R(t) by the
+5
+
+Wiener-Khinchine theorem,
+K(ω) = ∑
+i
+c2
+i
+ω2
+i
+δ(ω −ωi).
+(10)
+Here ωi are the eigenfrequencies of the lattice Hessian DQQ, ci are the coupling constants between
+the surface degree of freedom and ith normal mode, ci = ∑j D( j)
+PQVji where V is a matrix of with the
+eigenvectors of DQQ as columns. Note that we ignore the negative frequency components of the
+Fourier transform in Eq.10 as they are simply a reflection of the positive components. Eq.10 reveals
+that the peaks of power spectrum are nothing more than the phonon frequencies of the lattice
+weighted by their relative coupling to the site of interest. We will use this powerful interpretation
+throughout the rest of our paper.
+2.2
+Correlation function method
+The advantage of the method outlined in the previous section lies in its simplicity and interpretabil-
+ity, as it only requires computing and diagonalizing the mass-weighted Hessian of the lattice, DQQ.
+Its disadvantage is in the assumption of a PES of purely harmonic form (Eq.3), which limits its ap-
+plicability in systems where anharmonicities are significant, such solvated surfaces or surfaces with
+defects. An alternative approach, often used in the application of the GLE to liquid solutions, takes
+advantage of the fact that the random force R(t) must be uncorrelated with the system’s momenta:
+⟨R(t)p(0)⟩ = 0. This identity can be considered prerequisite for R(t) to be properly interpreted as
+a "random" noise, and is indeed consistent with Eq.9. Thus, by taking the time correlation function
+of both sides of Eq. 1 with the initial momentum we find,
+⟨ ˙p(t)p(0)⟩+
+�dW
+dq (t)p(0)
+�
+= −
+� t
+0 K(t −τ)⟨p(τ)p(0)⟩dτ.
+(11)
+Using MD simulation the force-momentum correlation functions (second term on the left-hand
+side) and the momentum autocorrelation function may be computed, and subsequently Eq.11 may
+be solved to find K(t). Unfortunately, solving Eq.11 with a high degree of numerical accuracy
+requires methods more complex than simply applying the fast Fourier transform and convolution
+6
+
+theorem.26,28 Details for methods used in this paper to solve Eq.11 may be found in the Appendix.
+2.3
+Extended Variable Transformation
+Computing the memory integral in Eq.1 is computationally intensive especially for systems with
+long memory decay rates. To circumvent this issue, it is common to expand the GLE back into a
+set of Markovian equations describing a system bilinearly coupled to bath of dissipative, stochastic
+harmonic oscillators. While in many ways this procedure is essentially the reverse of that presented
+in Section 2.1, using a dissipative HO bath is advantageous to using an energy conserving bath as
+the action of a continuum of energy conserving oscillators can often be represented with only one
+or two dissipative oscillations, greatly reducing the dimensionality of the equations of motion.
+Here we briefly summarize the method, excellent reviews can be found in Ref. 29 and Ref. 30.
+Given a GLE with a memory kernel that is a finite sum of exponentially damped sinusoids,
+K(t) =
+N
+∑
+i=1
+e−γit (Ci cos(ωit)+Di sin(ωit)),
+(12)
+the original non-Markovian equation of motion can be replaced with,
+d
+dt
+�
+�
+�
+p
+b
+�
+�
+� =
+�
+�
+�
+−dW
+dq
+0
+�
+�
+�+
+�
+�
+�
+0
+Apb
+Abp
+Ab
+�
+�
+�
+�
+�
+�
+p
+b
+�
+�
+�+
+�
+�
+�
+0
+0
+0
+Bb
+�
+�
+�
+�
+�
+�
+dW
+�
+�
+�.
+(13)
+Here p is the system’s momenta, and b is a set of bath variables we must involve in time with our
+system. dW is an array of uncorrelated Gaussian random variables satisfying
+�
+dWi(t)dWj(0)
+�
+=
+δi jδ(t), where δij is the Kronecker delta and δ(t) the Dirac delta. The matrix Ab is block diagonal
+with entries,
+Ab =
+�
+�
+�
+2γi
+�
+γ2
+i +ω2
+i
+−
+�
+γ2
+i +ω2
+i
+0
+�
+�
+�,
+(14)
+7
+
+and Apb and Abp are arrays of form,
+Apb =
+��
+Ci
+2 −2Diω2
+i
+γi
+�
+Ci
+2 +2Diω2
+i
+γi
+�
+,
+Abp =
+�
+�
+�
+�
+�
+Ci
+2 −2Diω2
+i
+γi
+�
+Ci
+2 +2Diω2
+i
+γi
+�
+�
+�
+�.
+(15)
+The matrix Bb is related to Ab by the equation,
+BbBT
+b = kBT(AQ +AT
+Q),
+(16)
+which ensures that the ensuing dynamics obey the fluctuation-dissipation theorem.
+Tully’s GLO model is nothing more than the N = 1 case of Eq.12, meanwhile in this paper, we
+determine the optimal number of terms N to use by analyzing K(t) calculated using the methods
+discussed previously in this section. In the remainder of this work, without loss of generality, we
+drop the sin terms in Eq.12, such that the memory kernel is simply a sum of exponentially damped
+cosines and the power spectrum is of Lorentzian form,
+K(ω) =
+N
+∑
+i=1
+Ci
+�
+γi
+γ2
+i +(ω −ωi)2
+�
+.
+(17)
+3
+Simulation Details
+Simulations using Effective Medium Theory (EMT) or Embedded Atom Method (EAM) force-
+fields for metal dynamics were performed using the Atomic Simulation Environment.31 The pa-
+rameters for these forcefields were taken from Ref. 32 and Ref. 33 respectively. Simulations using
+Lennard-Jones forcefield, both solvated and in vacuum state, were performed using LAMMPS.
+Lennard-Jones forcefield parameters were taken from Ref. 34. The solvent used was SPC/E.35
+All simulations were performed in two steps. First, a temperature equilibriation step was run
+for 50 picoseconds at 300K using a Langevin thermostat. Afterwards simulations were run in an
+constant energy ensemble using the velocity Verlet algorithm for 4 nanoseconds. Only data from
+8
+
+the NVE step was used in subsequent analysis and calculations. The bottom four corners of the
+lattice were rigidly constrained in order to remove any center of mass motion.
+For the surface scattering simulations used to generate data for Section 5, 5000 independent
+trajectories were averaged per value of the incident velocity to obtain sticking coefficients for
+GLE simulations, while 2000 independent trajectories were averaged for EMT simulations. For
+the surface desorption simulations, 2000 independent simulations were run in parallel, and the
+desorption rate constant was calculated by computing the steady-state flux over the barrier.
+4
+Memory kernels and power spectra for metal lattices
+We begin by analyzing the memory kernel for the fluctuations of a single atom site in the surface
+of 4x4x4 unit cell of Pt(111). Results were calculated for each surface site individually and sub-
+sequently averaged together. All memory kernels and power spectra presented in the main text
+are calculated via the CF approach. We present results using the PO approach in the supplemen-
+tary information, and will refer these results when necessary in the main text. Evidence for the
+convergence of the memory kernels presented here is given in Fig.S1 and Fig.S2.
+Figure 1A presents the three x, y, and z components of the memory kernel respectively calcu-
+lated from simulations using an Effective Medium Theory (EMT) forcefield. The x and y compo-
+nents arise from fluctuations in the plane of the lattice and the z component arises from fluctuations
+normal to it. Note the anisotropy between the x and y components and the z component, a simple
+consequence of the difference in the number of nearest-neighbors for surface sites. For the remain-
+der of the paper, we will focus only on Kz for the purposes of brevity and due to the fact that it is
+the coordinate which most effects the adsorption of reagent on the surface.
+In Figure 1B we present the noise power spectrum (Fourier transform of the memory kernel)
+of the zth component Kz(ν) specifically. As elaborated upon in Section 2, each peak in the power
+spectrum gives information about the lattice phonon modes and how they couple to the motion
+of a surface site. The center of the peak ν gives the frequency of the mode, the width γ gives
+9
+
+0
+5
+10
+15
+20
+t (ps)
+�100
+�50
+0
+50
+100
+150
+200
+K(t)
+X
+Y
+Z
+(A)
+0
+50
+100
+150
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+1200
+Kz(⌫)
+(B)
+ 
+ 
+ 
+νD =
+156 cm−1
+     
+  
+     
+ν = 131 cm−1 ω = 24.8 ps−1
+C = 146.3 ps−1 γ = 3.42 ps−1
+     
+  
+    
+ν = 18.6 cm−1 ω = 3.51 ps−1
+C = 38.0 ps−1 γ = 0.027 ps−1
+Figure 1: Memory kernel and random force power spectrum for surface sites of a Pt(111) lattice
+computed using an EMT forcefield. (A) Memory kernel for fluctuations in x/y (in-surface plane)
+and z (out of plane) directions. (B) Power spectrum of z component of the memory kernel. Red
+and blue lines are two Lorentzian functions optimized to fit the computed power spectrum (grey
+line). The grey dashed vertical line corresponds to the experimental Debye frequency. The inlets
+show the motion of the normal modes most associated with the red and blue lines as well as the
+parameters of the Lorentzians.
+the timescale of energy exchange or dissipation between the mode and the surface site, and the
+coefficient C represents the coupling strength.
+The power spectrum in Figure 1B is of a bimodal form. The blue peak - henceforth called the
+acoustic peak - is centered at a low frequency (ν = 18.6cm−1) and thus exchanges energy quite
+slowly (γ = 0.027ps−1), while the red term - henceforth called the Debye peak - is centered near
+the Debye frequency of Pt (ν = 131cm−1) and exchanges energy much faster (γ = 3.42ps−1). By
+comparing the power spectrum computing via the CF method to the power spectrum from the
+PO method (Figure S3), it is possible to determine precisely which normal modes of the lattice
+are primarily responsible for these two peaks. These normal modes are illustrated in the inlets in
+Figure 1B. Naturally, the acoustic peak arises from a longitudinal acoustic oscillations normal to
+the surface plane. Meanwhile, the Debye peak arises from many closely spaced normal modes
+near the Debye frequency, which are composed of local oscillations at a very small wavelength.
+In order to ensure the validity and transferability of our results, we tested the forcefields other
+than EMT. These results are illustrated in (Figure 2). The Lennard-Jones (LJ) model is based on a
+very different underlying physics than the EMT/EAM models (LJ model uses only pairwise inter-
+10
+
+0
+5
+10
+15
+20
+t (ps)
+0
+100
+200
+Kz(t)
+EMT
+EAM
+LJ
+(A)
+0
+50
+100
+150
+⌫ (cm�1)
+0
+1000
+2000
+3000
+Kz(⌫)
+EMT
+EAM
+LJ
+(B)
+Figure 2: (A) Memory kernel and (B) power spectrum for surface site fluctuations of Pt(111) simu-
+lated using three different atomistic models: Effective Medium Theory, Embedded-Atom Method,
+and a Lennard-Jones model.
+actions, while EMT/EAM are both many-body potentials based on the local atom density). Despite
+this fact, all three models produce the same qualitative bimodal form. Much of the quantitative dif-
+ference between the EMT/EAM and LJ models can be explained by the fact that the LJ model
+produces a lattice which is much more stiff compared to EMT and EAM. The lattice stiffness can
+be roughly quantified in terms of the average value of the mass-weighted Hessian klat = ⟨D2⟩. For
+EMT the stiffness of the 4x4x4 Pt(111) lattice is 19.5 kJ/(mol nm2), for EAM it is 19.8 kJ/(mol
+nm2), and for LJ it is 37.1 kJ/(mol nm2), corroborating the results in Figure 2.
+We also tested lattices of different elemental composition and surface structure (Figure S5).
+Once again, although variations were observed in the location, widths, and heights of the primary
+peaks of the power spectrum, all of the lattices exhibited the same qualitative bimodal response. In
+some ways, the universality here can be considered a simple consequence of Eq. 10. Essentially,
+there is a trade-off between the amplitude of each mode, which decreases as the inverse square
+11
+
+of the frequency, and the density of modes, which increases sharply near the Debye frequency.
+However, it is worth noting that this universality is not trivial. The bimodal behavior is not recov-
+ered in simple 1D systems with nearest-neighbor interaction (see Ref. 36 and 37 and Section S3
+of the Supplementary Material for more details), and therefore is an emergent property of the 3D
+metal lattice. Understanding the underlying physical properties of the 3D lattice responsible for
+this emergent behavior is an interesting direction we intend to elaborate on in a future publication.
+4.1
+Finite-size effects
+Increasing the size of the lattice should shift the frequencies of the acoustic modes in accordance
+to a change in the boundary conditions, which should in turn affect the memory kernel. In Figure
+3A/B we demonstrate that this is indeed the case: the acoustic peak shifts down in frequency as one
+increases the size of the lattice, while the Debye peak remains unchanged. Even when increasing
+the lattice size to as many as 8000 atoms, the memory kernel and power spectrum do not converge.
+In fact, as shown in the inlet of Figure 3B the frequency ratio between the acoustic peak of different
+size lattices roughly agrees with the results of an isotropic wave equation, suggesting that the
+acoustic peak will never converge to a fixed frequency, but rather decrease like as 1/l where l is
+the side-length of the lattice.
+Figure 3. encompasses one of the central and most important results of this paper: phonon-
+induced surface-site fluctuations depend strongly on the lattice size. This result has several rami-
+fications. First, it suggests than in the macroscopic limit (i.e. when the size of the lattice reaches
+∼ 1023 atoms, the frequencies of acoustic mode will be much too slow to affect any reaction dy-
+namics at the surface. In other words, the acoustic mode will be effectively frozen. Therefore,
+for observables that depend on memory (most notably rate constants), this result suggests that all
+finite-size simulations contain an intrinsic error which is purely kinetic in nature. We will demon-
+strate this explicitly in Section 5.
+The size-dependency of the memory kernel may also have ramifications for nanoparticle cata-
+lyst design, as it shows nanoparticle vibrational modes should behave quite differently than their
+12
+
+0
+10
+20
+30
+40
+50
+t (ps)
+�50
+0
+50
+100
+150
+200
+250
+Kz(t)
+0
+50
+100
+150
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+Kz(⌫)
+4x4x4
+6x6x6
+8x8x8
+10x10x10
+14x14x14
+20x20x20
+0
+50
+100
+150
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+Kz(⌫)
+4x4x4
+6x6x6
+8x8x8
+10x10x10
+14x14x14
+20x20x20
+(A)
+(B)
+4
+6
+8
+10
+14
+20
+Cell Side Length (# of Atoms)
+2
+4
+6
+8
+Ratio of Frequencies
+Simulation Results
+Isotropic Wave Equation
+Figure 3: (A) Memory kernel and (B) power spectra for surface site fluctuations of Pt(111) lattices
+of different sizes. The inlet in (B) illustrates a comparison the relative frequencies of the acoustic
+peak of the power spectra and the 1
+l scaling of an isotropic wave-equation with periodic boundary
+conditions.
+macroscopic counter parts, much as their electronic modes do. Indeed, experimental studies of
+electron relaxation in metal supported nanoparticles have already shown that the phonon-mediated
+dissipation of electron energy depends strongly on the nanoparticle size.38 While the system of in-
+terest in this study is the surface nuclei, not the electrons, we believe both results can be understood
+as a consequence of the same underlying phonon confinement effects.
+4.2
+Solvation Effects
+All of the results discussed so far have assumed a lattice in gas phase, under conditions of suffi-
+ciently low pressure and surface coverage. Such systems are highly idealized, as most catalysts
+operate under conditions of fairly high pressure and surface coverage and can often be solvated.
+Here we will explore how solvation affects surface site fluctuations by computing memory kernels
+for Pt(111) surfaces solvated in SPC/E water using the CF approach. We save the more difficult,
+yet still very important question, of surface coverage for future publication.
+Figure 4. demonstrates the difference between Pt(111) surfaces in vacuum versus in solvent.
+The primary difference comes in a damping of the acoustic mode, whose coupling to the surface
+site motion is much smaller when the surface is solvated. This effect is likely attributable to the
+13
+
+0
+50
+100
+150
+⌫ / cm�1
+0
+1000
+2000
+3000
+K(⌫)
+Vacuum
+Solvated
+0
+10
+20
+30
+40
+50
+t / ps
+0
+100
+200
+Kz(t)
+Vacuum
+Solvent
+(A)
+(B)
+Figure 4: (A) Memory kernel and (B) noise power spectra for surface site fluctuations of Pt(111)
+simulated using a LJ model with and without SPC/E solvent.
+additional pressure exerted by the solvent, making large fluctuations in the direction normal to
+the surface plane more energetically costly. The damping of the acoustic mode suggests that the
+finite-size effects discussed previously are likely far less important for solvated surfaces than they
+are for surfaces in gas-phase.
+5
+How memory affects adsorption and desorption
+Molecular beam scattering experiments are an invaluable tool for understanding the properties of
+surface reactions, elucidating information about binding potential energy surface and energy dissi-
+pation rates of the lattice.39–41 When the incident particles are sampled from an appropriate thermal
+distribution, the surface sticking probability can be shown to be proportional to the adsorption rate
+constant. Tully’s GLO model is often used in simulations of surface scattering (either reactive or
+non-reactive) as a cheap computational method for describing energy loss to the lattice during the
+14
+
+scattering process.17–21 In this section, we employ the LGLE for the same purpose, specifically
+studying the differences between the finite-size limit and the macroscopic limit (when the acoustic
+modes are held fixed).
+Most surface reactions are very complex, involving multiple steps coupling phonon and elec-
+tron modes together, sometimes non-adiabatically,42 and determining an accurate reaction PES
+from quantum chemistry calculations is itself a non-trivial challenge.43,44 To avoid such complex-
+ities so that we can focus on the role of phonons, here we study only the non-reactive scattering of
+Argon on Pt(111), and only in the direction normal to the surface. The PES is taken to be of Morse
+form,
+U(z) = D(1−e−a(z−z0))2,
+(18)
+where D a parameter which controls the depth of the reaction well, and a a parameter which
+controls the width of the well. The values of these parameters are taken from DFT calculations
+presented in Ref. 45 using a van der Waals density functional (vdW-DF2) and are presented in
+Table S1.
+0
+1
+2
+3
+KE(t = 0)/D
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+S
+GLE - Macrscopic Limit
+GLE - 2 Term
+GLE - 5 Term
+0
+1
+2
+3
+KE(t = 0)/D
+0.0
+0.2
+0.4
+0.6
+0.8
+S
+GLE - Macrscopic Limit
+GLE - 2 Term
+GLE - 5 Term
+EMT
+(A)
+(B)
+Figure 5: Sticking probabilities S as a function of the ratio of the incident kinetic energy to the
+well depth KE(t = 0)/D. (A) Results for Morse PES with D = 6.62eV. (B) Results for Morse PES
+with an increased well-depth, D = 30.62eV.
+Figure 5A illustrates variations the sticking probability for using 4 models for the metal phonons.
+The blue curve uses EMT to treat the metal degrees of freedom. The red and orange curves use
+15
+
+the LGLE (Eq.1) parameterized from a 4x4x4 EMT simulation to treat the metal. The red curve
+uses only uses only two damped sinusoids to fit K(t), while the orange curve uses a five term fit
+give a more accurate estimation of the memory kernel and power spectrum (see Figure S7.). The
+black curve corresponds to the extrapolated macroscopic limit of the LGLE, wherein the surface
+site motion is coupled to only to the Debye mode.
+The blue, red, and orange curves of Figure 5A largely agree with one another, illustrating
+that the LGLE accurately captures the dynamics of the forcefield it is parameterized from. More
+interesting however, is the consistent increase in the sticking probability between the nanoscale
+lattices (either modeled with EMT or GLE) and the macroscopic limit. This discrepancy can be
+qualitatively explained by the relative dissipation rates of the acoustic and Debye modes. Since
+nanoscale lattices couple the motion of surface atoms to the acoustic modes, and these acoustic
+mode dissipates energy much slower than the Debye mode, collisions with nanoscale lattices are
+more elastic. This effect can be observed more explicitly by studying histograms of the energy
+dissipated over many scattering trajectories (Figure S8).
+In Figure 5B we study the scattering probability but now with a well-depth that is nearly 5
+times greater. Increasing D increases the effective coupling between the adsorbate and it’s phonon
+bath, exacerbating the finite-size effects seen in Figure 5B.
+The implication of Figure 5 is that all nanoscale atomistic simulations of surface scattering
+contain an intrinsic error. This error is due purely to the phonon confinement effects placed by
+the boundary conditions, and can be exacerbated by errors in the adsorbate’s binding energy to the
+metal. Ideally, one would like to develop a quantitative formula for predicting the size of this error
+given a set of simple parameters for the phonon power spectrum and metal-adsorbate interaction.
+We have developed such a theory, and will present in in a subsequent publication.
+5.1
+Barrier Crossing
+Barrier crossing simulations are effectively the inverse of the surface scattering simulations dis-
+cussed in the previous section. Here we begin an ensemble of trajectories within the reactant well
+16
+
+and measure the average flux out of the well.
+2
+4
+6
+8
+10
+D (eV)
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+kd (ns�1)
+GLE - Macrscopic Limit
+GLE - 2 Term
+GLE - 5 Term
+Figure 6: Desorption rate constants kd as a function of well-depth D.
+In contrast to our simulations of surface scattering, our simulations of barrier crossing exhibit
+only a very minor difference between the results for the nanoscale GLE models and the extrapo-
+lated macroscopic limit (Figure 5). One possible explanation is the difference in initial conditions
+between scattering and barrier crossing simulations. In the scattering simulations the adsorbate
+begins in a non-equilibrium state and we observe it’s relaxation, meanwhile in barrier crossing
+simulations the adsorbate begins near the equilibrium state and the desorption rate constant arises
+from fluctuations out of that state.
+To those familiar with developments in reaction rate theory over the past 50+ years, the results
+of Figure 6 are somewhat surprising. Indeed the works of Kramers,46 Grote and Hynes,47 and
+others48,49 have all suggested that rate constants are deeply connected by the frequency of kicks
+from the bath, often in ways that cannot be captured by uni-dimensional transition state theory.
+However, such theories generally apply the GLE directly to the reaction coordinate, not to the
+reaction site, highlighting the necessity of relating the two together in order to develop quantitative
+theories for how phonon-induced memory affects reaction dynamics.
+17
+
+6
+Conclusions
+In this paper we presented the lattice generalized Langevin equation, a model for simulating the
+effects of lattice vibrations on surface atoms. The most important parameter in this model is the
+memory kernel. We parameterize the memory kernel using data from MD simulations, showing
+that it has a universal bimodal form due to coupling to both acoustic oscillations as well as modes
+near the Debye frequency. This bimodal form is non-trivial, as it is not recovered in simple 1D sys-
+tems with nearest neighbor interactions. Since the frequency of the acoustic oscillations depends
+on the size of the lattice, and nanoscale MD simulations impose unphysical phonon confinement
+effects, it is reasonable to assume that observables which depend surface phonons will also contain
+artifacts. We showed that that this was indeed the case for the surface trapping probability for a
+simple system of Argon on Pt(111). Interestingly however, the surface desorption rate for the same
+system exhibited very little dependence of the nature of the lattice memory kernel.
+The advantages of the LGLE model are, first, it’s computational efficiency, as it reduces the the
+N degrees of freedom of the lattice, to only a small handful on terms needed to describe the motion
+of a surface site. Second, the insight that can be gained from studying the memory kernel, as we
+illustrated throughout this paper. Third, the transferability of the model. Once the LGLE is pa-
+rameterized for a given lattice, any surface reaction with that lattice can use the same LGLE, given
+that the thermodynamic conditions (temperature/pressure/surface coverage/solvation) are roughly
+the same.
+Appendix 1. Methods for solving the Volterra equation
+Let us rewrite Eq.11 in the form,
+Cf (t) = −
+� t
+0 K(t −τ)Cp(τ)dτ,
+(19)
+18
+
+where Cf (t) is the force-momentum correlation function and Cp(t) the momentum autocorrelation
+function. Solving this equation via Fast Fourier Transform introduces numerical artifacts when
+Cp(ω) is near zero. Therefore it is often advantageous to solve Eq.19 in the time domain by
+discretizing the memory integral. Here we illustrate one such approach, which we used in this
+paper.
+To begin we take the derivative of Eq.19. Doing so gives an equation which is better defined
+and more numerically stable for small times t,
+˙Cf (t) = −K(t)Cp(0)−
+� t
+0 K(t −τ) ˙Cp(τ)dτ.
+(20)
+We then introduce a trapezoidal quadrature in order to evaluate the memory integral at discrete
+timesteps ∆t,
+˙Cf (t = 0) = −K(0)Cp(0),
+˙Cf (t = ∆t) = −K(∆t)Cp(0)− ∆t
+2
+�
+K(∆t) ˙Cp(0)+K(0) ˙Cp(∆t)
+�
+,
+˙Cf (t = N∆t) = −K(N∆t)Cp(0)− ∆t
+2
+�
+K(N∆t) ˙Cp(0)+K(0) ˙Cp(N∆t)
+�
+−��t
+N−1
+∑
+n=1
+K((N −n)∆t) ˙Cp(n∆t).
+(21)
+Equation 21 can be used to calculate K(t) with a recursive algorithm using ˙Cf (t), ˙Cp(t), and the
+values of K(t) at earlier time-steps.
+Supplementary Material
+See supplementary material for analysis of convergence of memory kernels using CF method, a
+comparison of memory kernels using CF and PO methods, memory kernels for other metal lat-
+tices other than Pt(111), memory kernels for 1D harmonic chains, and further details on scatter-
+ing/desorption simulations.
+19
+
+Data Availability
+Data that support the findings of this study are available from the corresponding author upon rea-
+sonable reque
+Acknowledgements
+AF and APW were supported by the Office of Science of the U.S. Department of Energy under
+Contract No. DE-SC0019441. This research used resources of the National Energy Research
+Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of
+Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. Ardavan
+Farahvash acknowledges support from the National Science Foundation Graduate Research Fel-
+lowship program.
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+25
+
+Supplementary information for "On using the generalized Langevin equation to
+model substrate phonons and their role in surface adsorption and desorption"
+Ardavan Farahvash,1 Mayank Agarwal,2 Andrew Peterson,2 and Adam P. Willard1, a)
+1)Department of Chemistry, Massachusetts Institute of Technology, Cambridge,
+Massachusetts 02139, USA
+2)Department of Chemical Engineering, Brown University, Providence,
+Rhode Island 02912, USA
+a)Electronic mail: [email protected]
+1
+arXiv:2301.04873v1  [physics.chem-ph]  12 Jan 2023
+
+SI.
+CONVERGENCE OF CF MEMORY KERNEL
+We have measured the convergence of the memory kernel and force power spectra both in terms
+of the step size (stride) between simulation snapshots used to compute the time correlation func-
+tions in Eq.9 and in terms of the total length the simulation used to compute the time correlation
+functions T. In Figure S1 we illustrate the simulation length convergence. We see that the short
+time (< 10ps or > 30cm−1) statistics converge very quickly with respect to simulation length.
+The long time, low frequency statistics however are slower to converge, especially in terms of
+the heights of the associated peaks. In particular, using simulation lengths less than 2ns seems to
+produce artificial oscillations at a very low frequency (∼ 3cm-1) in the memory kernel.
+S0
+0
+20
+40
+60
+80
+100
+120
+140
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+Kz(⌫)
+T = 1.0 ns
+T = 2.0 ns
+T = 3.0 ns
+T = 3.5 ns
+T = 4.0 ns
+0
+20
+40
+60
+80
+100
+t (ps)
+�50
+0
+50
+100
+150
+Kz(t)
+T = 1.0 ns
+T = 2.0 ns
+T = 3.0 ns
+T = 3.5 ns
+T = 4.0 ns
+(A)
+(B)
+FIG. S1. Convergence of memory kernel (A) and power spectra (B) for 4x4x4 Pt(111) lattice taken as a
+function of the simulation length T
+S1
+0
+20
+40
+60
+80
+100
+120
+140
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+1200
+Kz(⌫)
+stride = 5 fs
+stride = 10 fs
+stride = 25 fs
+stride = 50 fs
+0
+5
+10
+15
+20
+25
+30
+t (ps)
+�50
+0
+50
+100
+150
+Kz(t)
+stride = 5 fs
+stride = 10 fs
+stride = 25 fs
+stride = 50 fs
+(A)
+(B)
+FIG. S2. Convergence of memory kernel (A) and power spectra (B) for 4x4x4 Pt(111) lattice taken as a
+function of the step length.
+In terms of stride, Figure S2 shows that the memory kernel is very well converged using step
+size of 10fs between snapshots.
+2
+
+SII.
+COMPARISON OF MEMORY KERNELS USING PO AND CF TECHNIQUES
+In Figure S3 we compare the memory kernel and power spectra computed using correlation
+function method and projection operator method detailed in Section 2 of the main text. Two dif-
+ferences of note are the small frequency shift between the two methods, and that the CF memory
+kernel is much smoother. As we verified in Figure S1 and S2 that the memory kernel is well con-
+verged, the difference between the PO curves and CF curves in Figure S3 can only be attributed to
+small anharmonicities in the EMT forcefield. Despite some differences between the two methods,
+we see that the PO method gives the same bimodal behavior as the CF approach.
+0
+5
+10
+15
+20
+t (ps)
+�50
+0
+50
+100
+150
+Kz(t)
+CF Method
+PO Method
+0
+20
+40
+60
+80
+100
+120
+140
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+Kz(⌫)
+CF Method
+PO Method
+(A)
+(B)
+FIG. S3. Comparison of memory kernel (A) and power spectra (B) using CF and PO methods for 4x4x4
+Pt(111) lattice. The δ function form of the PO spectral density is represented with vertical lines.
+SIII.
+MACROSCOPIC LIMIT OF 1D HARMONIC CHAIN
+Consider a 1D chain of harmonic oscillators with spring constant k = mω2 and periodic bound-
+ary conditions. Every site in the chain is identical, and the dynamical matrix is given by,
+D2 = ω2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+−2 1
+0
+0
+... −1
+−1 2 −1
+0
+...
+0
+...
+...
+...
+...
+0
+...
+0
+−1
+2
+−1
+−1 ...
+0
+0
+−1
+2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(S1)
+3
+
+Taking our system to be a single site in the lattice, the resulting bath projected matrix is given by
+D2
+QQ = ω2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+2
+−1
+0
+...
+0
+−1
+2
+−1 ...
+0
+...
+...
+...
+0
+... −1
+2
+−1
+0
+...
+0
+−1
+2
+�
+�
+�
+�
+�
+�
+�
+�
+�
+�
+.
+(S2)
+This matrix may be diagonalized analytically allowing one to find a solution to the memory kernel
+via Eq.5,
+K(t) = 4k
+N
+N
+∑
+n=1
+cos2(θn)cos(2ωt sin(θn)),
+(S3)
+where N is the total length of the chain and θn =
+nπ
+2(N+1). If we take the limit as N → ∞, we see
+that this sum converges to an integral,
+K(t) = 8πk
+� π/2
+0
+dθ cos2(θ)cos(2ωt sin(θ)).
+(S4)
+This integral has no closed form solution. However it can be expressed in terms of Bessel func-
+tions,
+K(t) = 4ω2
+π
+J1(2ωt)
+t
+,
+(S5)
+where J is a Bessel function of the first kind.
+0
+20
+40
+60
+80
+t
+�2
+0
+2
+4
+K(t)
+0
+1
+2
+3
+4
+!
+0
+50
+100
+150
+200
+250
+300
+K(!)
+N = 10
+N = 50
+N = 250
+N ! 1
+(A)
+(B)
+FIG. S4. Comparison of memory kernel (A) and power spectra (B) for single site fluctuations of a 1D
+harmonic chains of various lengths with periodic boundary.
+In Figure S4 we illustrate the size dependence of the memory kernel for a site in a 1D chain.
+Like the 3D lattices presented in the main text, there is a frequency shift as we move to increase
+4
+
+the size of the chain. However, the power spectra is not bimodal, but rather a continuous sum
+of many modes which decrease in amplitude as we approach the chain’s Debye frequency 2ω.
+Furthermore, the memory kernel also does not decay exponentially, but rather as 1
+t , perhaps a
+consequence of the well-known ergodicity breaking in such systems.
+SIV.
+RESULTS FOR OTHER METALS AND SURFACES
+In Figure S5 we illustrate the power spectra of metal surfaces other than Pt(111). The power
+spectra of Au(111) and Pt(110) (Figure S5C and Figure S5D) clearly have the same bimodal be-
+havior as Pt(111). The power spectra of Cu(111) (Figure S5B) appears to be missing the Debye
+mode, however the real-time memory kernel (Figure S5A) has the same characteristic fast decay
+followed by coherent oscillations which decay much slower. Comparing Figure S5A and Figure
+S5B suggests that surface sites still do couple to modes near the Debye frequency in Cu(111),
+however such modes are of such a high frequency and dissipate energy so quickly that they are
+either overdamped, or not properly resolved due to numerical errors when computing the correla-
+tion functions. This suggestion is corroborated by the fact that the experimental Debye frequency
+of Cu is nearly two times greater than that of Au, and roughly 50% greater than that of Pt.
+5
+
+0
+10
+20
+30
+40
+50
+t (ps)
+�100
+0
+100
+200
+300
+400
+Kz(t)
+4x4x4
+6x6x6
+8x8x8
+0
+50
+100
+150
+200
+250
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+1200
+Kz(⌫)
+4x4x4
+6x6x6
+8x8x8
+(A)
+(B)
+0
+50
+100
+150
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+1200
+Kz(⌫)
+4x4x4
+6x6x6
+8x8x8
+0
+50
+100
+150
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+Kz(⌫)
+4x4x4
+6x6x6
+8x8x8
+(C)
+(D)
+FIG. S5. (A) Memory kernel for surface sites of Cu(111) lattices of various sizes simulated with EMT.
+(B) Associated power spectra of Cu(111). (C) Power spectra for Au(111) lattices simulated with EMT. (D)
+Power spectra for Pt(100) lattices simulated with EMT.
+6
+
+SV.
+ARGON SCATTERING AND DESORPTION SIMULATIONS
+0
+5
+10
+15
+20
+25
+30
+z - Ang
+0
+10
+20
+30
+40
+50
+60
+U(z) - kBT
+0
+5
+10
+15
+20
+25
+30
+z - Ang
+0
+2
+4
+6
+8
+10
+12
+U(z) - kBT
+(A)
+(B)
+FIG. S6. Morse potentials used in main text. The red lines are the harmonic fits to the potential in the well.
+Figure S6 demonstrates the Morse potentials used in the main text to describe the interaction
+of Argon and a platinum surface. In Figure S6B we increase the depth of the well D, keeping the
+frequency ωz the same. The exact parameters of these potentials are shown in Table S1.
+TABLE SI. Morse potential parameters between Argon on Pt(111) surface used in main text. Taken from
+Ref X.
+D (eV)
+a (Å
+−1)
+ωz (cm−1)
+Ar
+6.62
+44
+0.83
+Ar - Deepwell
+30.62
+44
+0.39
+In Figure S7 we present the power spectra of the GLE models used to conduct the surface
+scattering and desorption simulations in the main text. The 5 term fit includes some of the more
+minor peaks of the power spectra not included in the 2 term fit. The macroscopic limit only
+includes the Debye peak.
+In Figure S8 we present histograms of the energy lost from the adsorbate to the lattice during
+the surface scattering simulations using the D = 30.62 eV potential at two different values of the
+incident velocity. At low incident velocities, the distribution is bimodal and asymmetric, how-
+ever as we increase the incident velocity, the distribution becomes increasingly Gaussian. The
+bimodality at low incident velocities is a consequence of some of the trajectories being trapped,
+and other escaping. Trapped trajectories interact for longer with the lattice and therefore dissipate
+more energy.
+7
+
+0
+50
+100
+150
+⌫ (cm�1)
+0
+200
+400
+600
+800
+1000
+1200
+Kz(⌫)
+Calculated K(t)
+2 term fit
+5 term fit
+Macroscopic Limit
+FIG. S7. Power spectra for a 4x4x4 Pt(111) lattice calculated using the CF method overlayed with a 2 term
+Lorentzian fits, 5 term Lorentzian fit, and the macrscopic limit corresponding to only fitting the region of
+the power spectra near the Debye frequency.
+�200
+�150
+�100
+�50
+�E
+0.000
+0.005
+0.010
+0.015
+0.020
+0.025
+P(�E)
+�75
+�50
+�25
+0
+25
+50
+�E
+0.00
+0.02
+0.04
+0.06
+0.08
+0.10
+0.12
+P(�E)
+GLE - Macrscopic Limit
+GLE - 5 term
+(A)
+(B)
+FIG. S8.
+Histograms for energy dissipated during scattering using D = 30.62 eV Morse potential. (A)
+Using trajectories with KE(t = 0)/D = 0.75. (B) Using trajectories with incident KE to well-depth ratio of
+KE(t = 0)/D = 2.5.
+Interestingly, when KE(t = 0)/D = 2.5 all the trajectories escape, however, by analyzing the
+histograms in Figure S8B we see can still see a signature of the finite-size effects discussed in the
+main text. The 5-term GLE model dissipates less energy than the macroscopic limit GLE model
+due to coupling to acoustic modes.
+8
+

-NE1T4oBgHgl3EQfUgOU/content/tmp_files/2301.03091v1.pdf.txt

@@ -0,0 +1,737 @@
+Astronomy & Astrophysics manuscript no. main
+©ESO 2023
+January 10, 2023
+Letter to the Editor
+New multiple AGN systems with sub-arcsec separation :
+confirmation of candidates selected via the novel GMP method
+A. Ciurlo1, F. Mannucci2, S. Yeh3, A. Amiri2, 4, S. Carniani5, C. Cicone6, G. Cresci2, R. Khatun6, E. Lusso2, 4, A.
+Marasco7, C. Marconcini2, 4, A. Marconi4, E. Nardini2, E. Pancino2, P. Rosati8, P. Severgnini9, M. Scialpi4, 2, G.
+Tozzi4, 2, G. Venturi10, 2, C. Vignali11, and M. Volonteri12
+1 Department of Physics and Astronomy, University of California Los Angeles, 430 Portola Plaza, Los Angeles, CA 90095, USA
+e-mail: [email protected]
+2 INAF, Osservatorio Astrofisico di Arcetri, largo E. Fermi 5, 50125 Firenze, Italy
+3 W. M Keck Observatory, 65-1120 Mamalahoa Highway, Kamuela, HI 96743, USA
+4 Dipartimento di Fisica e Astronomia, Università di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino (Firenze), Italy
+5 Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126, Pisa, Italy
+6 Institute of Theoretical Astrophysics, University of Oslo, P.O Box 1029, Blindern, 0315 Oslo, Norway
+7 INAF-Osservatorio Astronomico di Padova, Vicolo Osservatorio 5, Padova, Italia
+8 University of Ferrara, Department of Physics and Earth Sciences, Via G. Saragat, 21-44122 Ferrara, Italy
+9 INAF, Osservatorio Astronomico di Brera, Via Brera 28,20121 Milano, Italy
+10 Instituto de Astrofísica, Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile
+11 Physics and Astronomy Department "Augusto Righi", Università di Bologna, Via Gobetti 93/2, 40129 Bologna, Italy
+12 Institu d’Astrophysique de Paris, 98bis Bd Arago, 75014 Paris, France
+Submitted January 6, 2023
+ABSTRACT
+The existence of multiple active galactic nuclei (AGN) at small projected distances on the sky is due to either the presence of multiple,
+in-spiraling SMBHs, or to gravitational lensing of a single AGN. Both phenomena allow us to address important astrophysical and
+cosmological questions. However, few kpc-separation multiple AGN are currently known. Recently, the newly-developed Gaia Multi
+peak (GMP) method provided numerous new candidate members of these populations. We present spatially resolved, integral-field
+spectroscopy of a sample of four GMP-selected multiple AGNs candidates. In all of these systems, we detect two or more components
+with sub-arcsec separations. We find that two of the systems are dual AGNs, one is either an intrinsic triple or a lensed dual AGN,
+while the last system is a chance AGN/star alignment. Our observations double the number of confirmed multiple AGNs at projected
+separations below 7 kpc at z > 0.5, present the first detection of a possible triple AGN in a single galaxy at z > 0.5, and successfully
+test the GMP method as a novel technique to discover previously unknown multiple AGNs.
+Key words. Galaxies: active – quasars: general – quasars: emission lines
+1. Introduction
+All current cosmological models describe galaxy formation as a
+hierarchical process in which small galaxies merge to form larger
+systems. This process also applies to the supermassive black-
+holes (SMBHs) that co-evolve with the host galaxy (Begelman
+et al. 1980). Given the long merging timescale (∼1 Gyr, e.g
+Tremmel et al. 2017), a population of dual or multiple SMBHs
+must exist in many galaxies (Volonteri et al. 2003). SMBHs
+are expected to accrete material from the merging host galax-
+ies, producing dual or multiple luminous active galactic nu-
+clei (AGNs) in the same galaxy (Steinborn et al. 2016; Rosas-
+Guevara et al. 2019; Volonteri et al. 2022). For example, Volon-
+teri et al. (2022) estimate that at z > 2 more than 1% of the
+bright AGNs (Lbol >1043 erg/s) are expected to have a compan-
+ion within 10 kpc. The discovery of dual AGNs at kiloparsec-
+scale separation is therefore crucial to support the hierarchical
+formation model. Additionally, since dual AGNs are the precur-
+sors of a binary phase, they allow us to study the merging steps
+leading to the emission of gravitational waves (e.g. Colpi 2014).
+Several tens of dual AGNs at separations above 10–20 kpc
+are known (e.g. Lemon et al. 2019; Chen et al. 2022, among
+many others). However, very few dual-AGN at separations be-
+low ∼5 kpc –compatible with being in the same host galaxy–
+have been discovered so far. There is a shortage of known
+close systems especially at intermediate and high redshifts, when
+galaxy mergers are more common (see De Rosa et al. 2019 and
+Mannucci et al. 2022 and references therein). This lack is due to
+the relatively low efficiency of the current selection techniques
+for sub-arcsec separations systems (Rubinur et al. 2019). In par-
+ticular, only four systems with separations below 5 kpc have
+been confirmed at z>0.5 (Junkkarinen et al. 2001; Chen et al.
+2022; Mannucci et al. 2022, Glikman et al., in prep). The small
+number of currently known dual AGN systems prevents us from
+testing cosmological model predictions such as the fraction of
+dual systems over the total AGN population, their evolution with
+redshifts and their mass and luminosity ratios (Volonteri et al.
+2022, and references therein).
+Thanks to its high spatial resolution and full sky coverage,
+the Gaia satellite is revolutionizing the field (e.g Lemon et al.
+Article number, page 1 of 6
+arXiv:2301.03091v1  [astro-ph.GA]  8 Jan 2023
+
+A&A proofs: manuscript no. main
+2019; Shen et al. 2021; Chen et al. 2022; Lemon et al. 2022). In
+particular, the Gaia Multi-peak (GMP) method (Mannucci et al.
+2022) allows us to select large numbers of dual systems with sep-
+arations down to ∼0.15" by searching for multiple peaks in the
+light profile of the Gaia sources. Mannucci et al. (2022) tested
+the efficiency of this method on 31 GMP-selected systems with
+HST (archival images of 26 systems) and LBT (newly obtained
+high-resolution observations of five systems) images. All these
+systems show multiple compact sources with sub-arcsec resolu-
+tion, confirming that this novel technique can be extremely effi-
+cient in selecting a sample of quasi-stellar objects with multiple
+components.
+The GMP-identified systems can also be lensed, high-
+redshift AGN, that appear as multiple components with small
+spatial separations. Strongly lensed AGN are rare and unique
+tools for measuring the Hubble parameter (e.g. Wong 2018) and
+for investigating AGN feedback at high redshift (e.g. Feruglio
+et al. 2017; Tozzi et al. 2021). In particular, very compact sys-
+tems (sub-arcsec separations) allow us to investigate the mass
+distribution of lensing galaxies to a regime lower than what is
+typically probed by current galaxy-scale lenses surveys (e.g.,
+SLACS Bolton et al. 2008; Shajib et al. 2021). The sensitivity
+to such low-mass dark matter halos can be used to study the na-
+ture of dark matter (e.g. Casadio et al. 2021).
+A crucial next step is to understand the nature of the GMP-
+selected systems: intrinsically multiple AGNs, gravitationally-
+lensed systems or an AGN plus a foreground star. Integral
+field spectroscopy is particularly well-suited to extract spatially-
+resolved spectra of each component of these systems, thus
+helping us discriminate among these three scenarios. Here, we
+present the first spatially resolved spectroscopy of four GMP-
+selected systems, observed with the adaptive optics (AO) integral
+field spectrograph OSIRIS at W. M. Keck Observatory (Larkin
+et al. 2006). The goals of these observations are: (1) resolving
+point-sources in dual-AGN candidates to test the success rate
+of the GMP technique; (2) differentiating AGNs from stars in
+resolved systems, based on their spectral properties; (3) classify-
+ing the systems as intrinsically multiple vs. lensed AGNs, based
+on the differences between their spectra.
+This letter is structured as follows. Observations and data re-
+duction are reported in Section 2, the classification of each sys-
+tem is discussed in Section 3. Our conclusions are summarized
+in Section 4. All magnitudes we report are in Vega system and
+we used the cosmological parameters from Planck Collaboration
+et al. (2020).
+2. Target selection, observations, and data
+reduction
+Our targets were extracted from the Milliquas v7.2 catalog
+(Flesch 2021) by selecting systems observable from Keck, with
+spectroscopic redshifts z > 0.5, and redshift such as to have at
+least one bright line (Hα for all these systems) inside one of near-
+IR bands used by OSIRIS. All sources were selected through
+the GMP method by having values of ipd_ frac_multi_peak1
+above the threshold of 10 (Mannucci et al. 2022). We exclude ob-
+jects where clear stellar features at zero velocity in their archival
+ground-based spectrum reveals the presence of a chance align-
+ment between an AGN and a foreground star.
+All observations and observing conditions are reported in
+Table 1. We observed systems J1026+6023, J1608+2716 and
+J1613+1708 on March 19th 2022 with laser guide star (LGS)
+1 the parameter of the Gaia archive used for the GMP selection
+AO, with a 50 mas pixelscale. On our second scheduled observ-
+ing date, August 12th 2022, the laser was not available, so we
+observed system J2335+3201 with a natural guide star (NGS)
+correction instead. The tip and tilt star for this target is faint
+(14.33 magnitudes in R band, fainter than Keck’s nominal NGS
+limit), therefore the correction was worse than during our other
+observations. Given the lower spatial resolution provided by this
+correction, we opted for a larger pixelscale of 100 mas.
+Due to their relatively large separation (0.75" and 0.61", re-
+spectively), systems J1613+1708 and J2335+3201 are already
+resolved into two sources in the Gaia archive. This allows us to
+know the separation angle and the system orientation in advance.
+Therefore, we used the small OSIRIS field of view (0.8"×3.2" at
+50 mas platescale, 1.6"×6.4" at 100 mas platescale) which cor-
+responds to broad-band filters (respectively Hbb from 1.473 to
+1.803 µm and Jbb from 1.180 to 1.416 µm). The other two tar-
+gets (J1026+6023 and J1608+1716) appear as single entries in
+the Gaia archive. Therefore, we observed them with a larger field
+of view (1.6"×3.2") that allowed us to account for the unknown
+orientation of the systems but that comes with a narrower spec-
+tral coverage (respectively Hn5 from 1.721 to 1.808 µm and Kn5
+from 2.292 to 2.408 µm). In addition to the science targets, each
+night we also observed a standard star of spectral type A for tel-
+luric calibration, and a field of view free of targets for sky sub-
+traction. All data cubes were assembled and reduced using the
+standard OSIRIS pipeline (Lockhart et al. 2019).
+For each target, we extract the spectrum of all detected com-
+ponents by taking the weighted sum in the squared apertures
+shown in the Figure 1 (left panels). We calculate the weighting
+factor for each spaxel by extracting its corresponding spectrum
+and measuring the total Hα flux. In this way, the signal-to-noise
+is maximized while the cross-contamination between different
+components and the aperture size impact are minimized. We note
+that this technique applies because the sources are expected to be
+point-like and, therefore, to show no spectral variation across the
+field of view.
+3. Results
+We find that all four targets are resolved into multiple point-
+sources, with separations in the expected range (Mannucci et al.
+2022). The images and the spectra of all the systems are shown
+in Figure 1. These spatially-resolved spectra allow us to study
+the nature of each object, as summarized in Table 2.
+3.1. J1026+6023
+J1026+6023 is composed of an AGN and a star. The AGN shows
+a Hα line with broad and narrow components and a prominent
+narrow [NII]λ6584 line with a redshift of z=1.659. This AGN
+is at 0.61" separation from an object with a featureless spectrum
+which we identify as a foreground star. The AGN (component A)
+is the brightest object in the optical band, sampled by Gaia and
+the Sloan Digital Sky Survey (SDSS, Lyke et al. 2020), while
+the star is the brightest object in the near-IR H band sampled by
+the Keck spectra (component B). Chance AGN/star alignments
+of this kind are expected to be 30% of the GMP-selected targets
+(Mannucci et al. 2022).
+3.2. J1608+2716
+J1608+2716 is an obscured quasi-stellar object (QSO), at
+z=2.575, with AV ∼1.8 as estimated from the SDSS spectrum.
+Article number, page 2 of 6
+
+A. Ciurlo et al.: Unveiling multiple AGNs via the GMP method
+0.5
+0.0
+0.5
+0.5
+0.0
+0.5
+arcsec
+A
+B
+1.72
+1.74
+1.76
+1.78
+1.80
+0
+10
+20
+30
+Ha
+[NII]
+[NII]
+J1026+6023   z=1.659
+0.5
+0.0
+0.5
+0.4
+0.2
+0.0
+0.2
+0.4
+arcsec
+A
+B
+C
+2.30
+2.32
+2.34
+2.36
+2.38
+2.40
+0
+20
+40
+60
+Ha
+[NII]
+[NII]
+[SII]
+J1608+2716   z=2.578
+x 2
+x 4
+0.5
+0.0
+0.5
+0.50
+0.25
+0.00
+0.25
+0.50
+arcsec
+A
+B
+1.50
+1.55
+1.60
+1.65
+1.70
+1.75
+1.80
+0
+20
+40
+60
+HeI
+Ha
+[NII]
+J1613+1708   z=1.547
+1
+0
+1
+arcsec
+1.0
+0.5
+0.0
+0.5
+1.0
+arcsec
+A
+B
+1.18
+1.20
+1.22
+1.24
+1.26
+1.28
+1.30
+1.32
+Wavelength (mic)
+0
+50
+100
+150
+Ha
+[NII]
+J2335+3201   z=0.904
+x 10
+Fig. 1. Hα emission line maps (left) and spectra (right) of the systems observed with OSIRIS (target name and redshift reported in the right panels).
+The line maps are oriented with North up and West right. The spectra shown on the right panels have been extracted over the squared apertures
+marked on the left panes (with the same color-coding) . Each component of the systems is labelled as in Table 2. To optimize the visualization some
+of the spectra have been multiplied by the factors indicated in the labels. Vertical dotted lines show the position of the main expected emission
+lines.
+Target
+RA
+DEC
+PA
+IPDfmp
+Redshift
+Band
+Texp×Nexp
+FWHM
+seeing
+AO
+J1026+6023
+10:26:31.13
++60:23:30.13
+102◦
+21
+1.660
+Hn5
+900 s ×4
+0.10”
+0.7”
+LGS
+J1608+2716
+16:08:29.23
++27:16:26.74
+-357◦
+14
+2.575
+Kn5
+900 s ×6
+0.09”
+0.7”
+LGS
+J1613+1708
+16:13:20.01
++17:08:39.40
+135◦
+14
+1.547
+Hbb
+900 s ×4
+0.11”
+0.7”
+LGS
+J2335+3201
+23:35:22.52
++32:01:09.08
+-106◦
+13
+0.904
+Jbb
+600 s ×2
+0.42”
+0.9”
+NGS
+Table 1. Main properties of the four targets studied in this work, along with Keck OSIRIS observational setup. IPDfmp is the value of the
+ipd_frac_multi_peak parameter of the Gaia archive used for the GMP selection. Redshift are obtained from SDSS ground-based spectra, as
+reported in the Milliquas catalog. FWHMs are calculated on isolated sources. The seeing corresponds to the DIMM (Differential Image Motion
+Monitor) seeing mean value (at zenith, at 0.5 µm), as reported by the Maunakea Weather Center for the same night of the observations.
+Our observations reveal three components: a central brightest
+one (component A), one 0.25" to the east (component B) and
+one 0.29" towards north west (component C). Faint extensions
+are visible for components A and C, but their low luminosity,
+compared with nearby components, and the extended wings of
+the AO point-spread-function (PSF) do not allow us to extract
+independent spectra. Due to the shorter wavelength range used
+in the observations, the spectra only cover the broad Hα line
+and a limited part of the continuum on both sides. All the three
+components show broad Hα lines at similar redshifts, with veloc-
+ity dispersion of about 5500 km/sec full width at half maximum
+(FWHM), but with slightly offset line centers.
+There are three main possible explanations to a triple object:
+1) a triple lensed system, i.e, three images of the same object;
+2) lensing of a dual AGN: two distinct objects, one of which
+with two detected lensed images; 3) a systems of three different
+AGNs, a possibility predicted by current models (e.g Ni et al.
+2022; Bhowmick et al. 2020; Volonteri et al. 2022) and previ-
+ously observed in the Local Universe (e.g Foord et al. 2021; Ya-
+dav et al. 2021).
+To unveil the nature of this source we can consider the fol-
+lowing points:
+– Line position and profile: component B displays both a dif-
+ferent line profile and radial velocity with respect to the cen-
+tral, brightest component A, as shown in Figure 2. Gaussian
+fits to the emission lines of all components show that the
+Hα line of component B (in blue) is centered at lower wave-
+Article number, page 3 of 6
+
+A&A proofs: manuscript no. main
+Target
+Class
+Separation
+Line
+Center
+redshift
+arcsec
+kpc
+(µm)
+J1026+6023A
+AGN
+Hα
+1.7451
+1.659
+[NII]6854
+1.7503
+1.667
+J1026+6023B
+Star
+0.61
+-
+-
+-
+-
+J1608+2716A
+dual/triple AGN
+Hα+[NII]
+2.3524
+2.584
+J1608+2716B
+0.25
+2.0
+Hα+[NII]
+2.3467
+2.576
+J1608+2716C
+0.29
+2.4
+Hα+[NII]
+2.3527
+2.585
+J1613+1708A
+dual AGN
+Hα+[NII]
+1.6732
+1.550
+J1613+1708B
+0.71
+6.1
+Hα+[NII]
+1.6702
+1.545
+J2335+3201A
+dual AGN
+Hα+[NII]
+1.2492
+0.904
+J2335+3201B
+0.61
+4.8
+Hα+[NII]
+1.2508
+0.906
+Table 2. Summary of the results from our OSIRIS observations: most probable classification, projected angular and linear distances from the
+brightest object, and center of the observed lines.
+lengths, with a difference of ∼720 km/sec, and has a FWHM
+larger than component A by 1200 km/sec. We estimated the
+uncertainties on the center and the FWHM of the best-fit
+Gaussians by adding Gaussian noise to the spectra at the ob-
+served amplitude, and computing the fit again. This process
+was repeated 2000 times for each line. The distribution of
+the resulting centers and FWHM are show in Figure 2 (cen-
+ter and left panels). This shows that the differences in center
+and width between components A and B are highly signifi-
+cant. We can exclude spatially-dependent calibration issues
+because the sky lines in spectra extracted at the locations of
+the components overlap perfectly. In contrast to component
+B, component C has a spectrum compatible with A.
+– Variability and time lag: given the small projected separa-
+tion (0.25"), in the case of lensing, the time delay between
+components A and B is 2 days at most (Lieu 2008). For in-
+trinsic variability to be at the origin of the differences above,
+this timescale must be larger than (or of the same order of)
+the size of the broad-line emitting region (BLR). Bentz et al.
+(2013) have estimated the radius of the Balmer-line emitting
+part of the BLR as a function of the luminosity of the con-
+tinuum λL(λ) at 5100 Å. For J1608+2716, this luminosity
+–estimated from the SDSS spectrum and the G-band Gaia
+magnitude– is log(λ L(λ))/erg sec−1=46.0±0.2. For this lu-
+minosity, Bentz et al. (2013) estimate a radius of the BLR
+of ∼ 400 light-days. Even assuming that the luminosity of
+this object is boosted by a factor of 10 by lensing, the radius
+would be ∼ 100 light-days. This is much larger than the ex-
+pected delay. Therefore, in the case of lensing, no significant
+variability of the Hα line would be expected between the two
+images.
+– Lensing: component C (red in Figure 2) has center and
+FWHM compatible with the brightest component A. How-
+ever, the two lines have significantly different equivalent
+widths (377Å for component A, vs. 232Å for component C).
+This difference, in the lensing scenario, could be attributed to
+microlensing of the continuum by single stars in the lensing
+galaxy (e.g. Hutsemékers et al. 2010). If A and C are lensed
+images of the same QSO, the B image would be the sec-
+ond component of a dual AGN, however producing a single
+image if it lies outside the radial caustic of a general ellipti-
+cal mass distribution. In any case, a compact lensing galaxy
+should be present.
+– Missing lensing galaxy: nothing is detected in the observed
+spectra besides the QSOs and the faint extensions of com-
+ponent A and C. Two lensed images of a QSO at zs = 2.57
+separated by 0.25" (with a third image strongly demagnified
+near the center) can be obtained with a lens galaxy with a
+mass of M∼ 1010M⊙, by assuming it at redshift zL ∼ 0.5 − 1
+and by requiring the separation to be twice the Einstein ra-
+dius of a singular isothermal sphere2. Such a compact lensed
+system would only sample the central part of the lensing
+galaxy where the contribution of dark matter is gravitation-
+ally subdominant with respect to stellar mass, with a contri-
+bution lower than the uncertainties. Assuming that this mass
+is dominated by stars, we estimate a galaxy magnitude be-
+tween Ks∼19.2 at z=0.5 and Ks∼20.5 at z=1.0 (Longhetti &
+Saracco 2009, for an early-type galaxy with a Chabrier initial
+mass function). As a comparison, the QSO has Ks∼19.1, es-
+timated using Gaia magnitudes and SDSS spectra. A lensed
+galaxy at z=0.5 would, therefore, be easily detected also con-
+sidering that it is not a point source, while would be below
+detection at z=1, especially if it dust extincted. The nucleus
+of the lensing galaxy could be the faint extension of compo-
+nent A, that otherwise could be the QSO host galaxy.
+In conclusion, the differences in line center and profile be-
+tween components A and B, together with the small time delay
+between the images, show that this is not a single, triply-imaged
+lensed QSO, but that at least two components must be present.
+Components A and C are compatible with a double lens system
+with some contribution from microlensing, with the possible de-
+tection of the host galaxy. This system would be a lensed dual
+QSO, similar to the system described by Lemon et al. (2022).
+However, since a foreground lensing galaxy is not clearly de-
+tected, this system could also be a physically triple AGN. Some
+knowledge of the spectral energy distribution of the three sources
+would further help to understand the nature of this system.
+3.3. J1613+1708
+J1613+1708 is a very blue QSO, with no evidence for dust ex-
+tinction in the SDSS spectrum. We find that this system shows
+two components with similar luminosities and a separation of
+0.71" (6.1 kpc). A bright Hα line is present in both spectra,
+with a velocity shift of ∼ 500 km/sec, corresponding to red-
+shifts of z=1.550 and z=1.545 respectively. The line width are
+also very different: 6200 km/sec FWHM for component A, and
+3100 km/sec for component B. In case of lensing, given its lu-
+minosity at 5100 Å of log(λ L(λ)) = 45.2 ± 0.1, no significant
+variations of the Hα line are expected on timescales shorter than
+2 θS IS
+E
+= [DLS /(DLDs) 4GM/c2]1/2, where DL, DS are the angular di-
+ameter distances of the lens, the source and DLS the one between the
+lens and the source.
+Article number, page 4 of 6
+
+A. Ciurlo et al.: Unveiling multiple AGNs via the GMP method
+20
+40
+60
+0
+100
+200
+300
+Center
+0
+100
+200
+300
+Sigma
+0
+20
+40
+60
+0
+100
+200
+Center
+0
+100
+200
+300
+Sigma
+2.30
+2.32
+2.34
+2.36
+2.38
+2.40
+Wavelength [ m]
+20
+40
+60
+2.346
+2.348
+2.350
+2.352
+Wavelength [ m]
+0
+100
+200
+300
+Center
+5000
+5500
+6000
+6500
+FWHM [Km/sec]
+0
+100
+200
+Sigma
+J1608+2716
+Fig. 2. Comparison of the Hα lines of the three components of J1608+2716. From top to bottom: A, B and C components, color-coded as in
+Figure 1. Left panels: observed emission line (solid, thick line) and fit with a Gaussian profile plus a constant (think solid line). The center of
+the best-fitting Gaussian is reported as a vertical dashed line. In all panels the green dotted line show the fit to A (the brightest component) for
+comparison. Center and right columns: centroids (center) and FWHM (right) distributions determined by our Gaussian fit on 2000 stochastic
+realisations of the observed spectra, each obtained by injecting noise into the data.
+160 days (or 50 days assuming a lensing magnification by a fac-
+tor of 10 Bentz et al. 2013). In contrast, the delay expected due
+to the separation of the two components would be 10 days at
+most. As a consequence, we conclude that the two objects are
+associated with two different AGNs in a single host.
+3.4. J2335+3201
+This is a low-extinction (AV∼0.4, estimated from the SDSS spec-
+trum) system at z∼0.9 showing two distinct components 0.61"
+(4.8 kpc) away, with a large (∼ 12) luminosity ratio. We find that
+both objects show a broad Hα line width (FWHM=2900 km/sec
+for the component A and 2700 km/sec for component B). The
+two lines show a significant velocity shift of about 400 km/sec,
+and different line profiles. The system has log(λLλ)=44.9 at
+5100 Å, implying variability timescales of ∼100 days (30 days
+in case of a lensing magnification by a factor of 10), to be com-
+pared with the expected delay of 2 days. Therefore, also in this
+case, the differences are better explained by a dual AGN system.
+4. Conclusions
+We used AO-assisted, spatially-resolved spectroscopy to unveil
+the nature of four complex AGN systems at redshifts between
+0.9 and 2.4 selected through the GMP method. As expected
+by the GMP selection, all these objects show multiple com-
+ponents with sub-arcsec separations. Target J1026+6023 is
+better described by a AGN/star alignment (given the featureless
+continuum), while emission from broad lines typical of QSO are
+seen in all the components of the remaining three systems. Ve-
+locity shifts of a few hundreds km/sec are seen in J1608+2716,
+J1613+1708 and J2335+3201, compatible with being due to
+multiple distinct SMBHs likely to be in the process of merging
+inside a single host. The differences in line profiles and projected
+separations are indeed best reproduced by intrinsically distinct
+SMBHs rather than lensing by a foreground galaxy. In fact,
+the luminosity of the three QSOs, even allowing for possible
+lensing magnification, implying large sizes of the BLR and
+therefore slow variability on timescales of several tens/hundreds
+of days. Since the expected time delay between different lensed
+images would correspond to a few days at most, the differences
+cannot be due to lensing delay. Moreover, there is no evidence
+for a foreground lensing galaxy. These observations confirm
+that a sizeble sample of intrinsic multiple AGNs can be obtained
+with a reasonable amount of resolved spectra of GMP selected
+systems. Future observations from the ground (especially with
+VLT/MUSE, VLT/ERIS, and Keck/OSIRIS) and from the space
+(HST/STIS, JWST) will allow us to largely increase the number
+of confirmed multiple systems and begin to compare the results
+with theoretical predictions on galaxy formation and evolution.
+Acknowledgements. AC acknowledges support from NSF AAG grant AST-
+1412615, Jim and Lori Keir, the W. M. Keck Observatory Keck Visiting Scholar
+program, the Gordon and Betty Moore Foundation, the Heising-Simons Founda-
+tion, and Howard and Astrid Preston. GC, FM, AM and EN acknowledge support
+by INAF Large Grants "The metal circle: a new sharp view of the baryon cycle up
+to Cosmic Dawn with the latest generation IFU facilities" and “Dual and binary
+supermassive black holes in the multi-messenger era: from galaxy mergers to
+gravitational waves” (Bando Ricerca Fondamentale INAF 2022). GV acknowl-
+edges support from ANID program FONDECYT Postdoctorado 3200802. The
+authors wish to recognize and acknowledge the very significant cultural role and
+reverence that the summit of Maunakea has always had within the indigenous
+Hawaiian community. We are most fortunate to have the opportunity to conduct
+observations from this mountain.
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+

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+1
+Collaborative Semantic Communication at the Edge
+Wing Fei Lo, Nitish Mital, Member, IEEE, Haotian Wu, Graduate Student Member, IEEE,
+Deniz G¨und¨uz, Fellow, IEEE
+Abstract—We study the collaborative image retrieval problem
+at the wireless edge, where multiple edge devices capture images
+of the same object from different angles and locations, which
+are then used jointly to retrieve similar images at the edge
+server over a shared multiple access channel (MAC). We propose
+two novel deep learning-based joint source and channel coding
+(JSCC) schemes for the task over both additive white Gaussian
+noise (AWGN) and Rayleigh slow fading channels, with the aim
+of maximizing the retrieval accuracy under a total bandwidth
+constraint. The proposed schemes are evaluated on a wide
+range of channel signal-to-noise ratios (SNRs), and shown to
+outperform the single-device JSCC and the separation-based
+multiple-access benchmarks. We also propose two novel SNR-
+aware JSCC schemes with attention modules to improve the
+performance in the case of channel mismatch between training
+and test instances.
+Index Terms—Semantic communication, Internet of Things,
+person re-identification, deep joint source and channel coding,
+collaborative image retrieval
+I. INTRODUCTION
+I
+N recent years, machine learning tasks at the wireless edge
+have been studied extensively in the literature, including
+distributed inference problems over wireless channels [1]–[3].
+In distributed inference problems, it is often assumed that
+centrally trained models, e.g. deep neural networks (DNNs)
+are employed across multiple distributed nodes, which have
+limited communication resources. Communication is essential
+in scenarios in which data is available at different nodes, and
+exploiting this data can increase the inference accuracy. In
+particular, in image retrieval, image of an object or a person
+taken by an edge device is used to identify the images of the
+same object or person in a gallery database. The images in the
+database may be taken by different cameras, from different
+angles, and at different times, making re-identification (ReID)
+a highly challenging inference problem. Note that, unlike most
+conventional classification or regression problems, which can
+be carried out locally at the edge device, for image retrieval,
+remote inference is essential even if the edge devices have
+unlimited computational power, as the gallery database is only
+available at the edge server. On the other hand, due to latency
+and bandwidth constraints, sending the whole image via the
+wireless channel is not feasible. Instead, learning-based feature
+extraction is done at the edge, and only the most important
+features of the source image should be sent to the edge server
+through the wireless channel.
+In [4], both separation-based and joint source channel
+coding (JSCC) approaches have been studied for feature trans-
+mission in remote image retrieval. While Shannon’s separation
+The authors are with the Department of Electrical and Electronic Engi-
+neering, Imperial College London, London SW7 2AZ, U.K. (e-mail: hao-
+[email protected]).
+Fig. 1: Illustration of the two-device collaborative image
+retrieval problem at the wireless edge.
+theorem [5] states that separating source and channel coding
+can achieve asymptotic optimality, this theorem breaks down
+in finite block-lengths. We typically have much more stringent
+latency constraints on edge inference applications compared
+to the delivery of images or videos; hence, our interest is in
+the very short blocklengths, where the separation typically has
+very poor performance. An autoencoder-based JSCC (JSCC-
+AE) scheme is proposed in [4], and it is shown to outperform
+its digital counterpart under all channel conditions.
+In this paper, we study the collaborative ReID problem,
+where two edge devices capture images of the same object,
+which are then used to identify similar images in a gallery
+database at an edge server. The increasing number of edge
+devices raises new requirements for collaborative inference:
+edge devices must collaborate not only with the edge server,
+but also with each other as multiple images can provide addi-
+tional information and can potentially improve ReID accuracy.
+The goal of this paper is to develop a deep learning-based
+JSCC scheme for the two-device scenario, which maximizes
+the accuracy of the retrieval task while communicating over a
+shared multiple access channel (MAC). We first consider an
+orthogonal multiple access (OMA) scheme employing time
+division multiple access (TDMA) with distributed JSCC, and
+show that it outperforms the schemes in [4], as well as a
+conventional separate source-channel coding scheme, where
+each device transmits a quantized version of its features
+to the receiver using capacity-achieving channel codes. In
+addition, we study an alternative non-orthogonal multiple
+access (NOMA) approach. Benefits of NOMA transmission
+in various distributed inference and training problems have
+recently received significant interest [6]–[8]. In this approach,
+our goal is exploit the superposition property of the wireless
+medium, and the features transmitted as analog values over the
+shared wireless channel get aggregated “over-the-air” rather
+than interfering with each other. We evaluate these schemes on
+the additive white Gaussian noise (AWGN) and Rayleigh slow
+fading channels. Inspired by the attention mechanism in adap-
+tive JSCC [9]–[11], we also propose an SNR-aware scheme
+for the AWGN channel to adjust the networks depending on
+the SNRs.
+arXiv:2301.03996v1  [eess.IV]  10 Jan 2023
+
+()
+Pre-processing
+Edge device 1
+(α)
+Wireless
+ID
+channel
+prediction
+()
+Pedestrian
+Pre-processing
+Edge server
+Edge device 22
+Our main contributions can be summarized as follows:
+• To the best of our knowledge, this is the first paper
+to study collaborative inference among edge devices for
+joint retrieval. We propose two new collaborative JSCC
+schemes for OMA and NOMA transmissions, and show
+the superiority of the latter.
+• We construct and analyze DNN architectures for a chan-
+nel state information (CSI)-aware JSCC scheme (SNR-
+aware and channel fading-aware), where a single network
+is trained to exploit the channel state information for
+channel equalization and SNR-adaptation.
+II. RELATED WORK
+A. Image retrieval
+Image retrieval task aims to improve the quality of identity
+recognition, particularly in surveillance applications. Given a
+query image, an image retrieval model assesses its similarities
+with gallery images, and matches is to the ‘nearest’ ones.
+Performance can be evaluated through top-1 retrieval accuracy
+[12]. Image retrieval task has received significant attention in
+recent years [13] thanks to the tremendous success of deep
+learning technologies [14].
+B. Remote inference at the wireless edge
+Classical communication systems are designed to deliver
+source signals, such as images, audio, or video, to a re-
+ceiver with the highest end-to-end fidelity. However, with
+the rapid growth of machine intelligence and the associated
+machine-to-machine communications, the goal of emergent
+communication systems is shifting towards making accurate
+inferences about a remote signal rather than reconstructing
+it [1]. Literature on joint edge-device inference [15], [16]
+mostly focus on rate-limited scenario, and ignore the channel
+effects. Jankowski et al. proposed a retrieval-oriented image
+compression scheme and a JSCC scheme for the retrieval
+task [4] with state-of-the-art performance. Remote inference
+problems are also attracting significant interest in the context
+of the emerging semantic communication paradigm [17].
+C. Multi-device collaborative learning
+Existing multi-device algorithms mainly focus on image
+transmission [18] and classification tasks [19]. Shao et al.
+proposed a task-oriented communication scheme for multi-
+device collaborative edge inference [19], which utilizes the
+information bottleneck (IB) principle for feature extraction and
+the deterministic distributed information bottleneck (DDIB)
+principle for distributed feature encoding. Different from pre-
+vious work, our paper explores cooperation for the image
+retrieval task.
+III. SYSTEM MODEL
+We consider two transmitters, each of them having access
+to images of the same object taken by a different camera.
+We denote the image observed by transmitter i by si ∈ Rp,
+i = 1, 2. Transmitter i employs an encoding function Ei :
+Rp → Cq, where xi = Ei(si) ∈ Cq. Here, q represents the
+available channel bandwidth, and r ≜ q
+p is the bandwidth ratio.
+The decoder function D : Cq → D is employed at the receiver,
+where D ≡ {1, 2, . . . , D}, and D is the size of the database,
+maps the received signal y to the result of the retrieval task.
+Channel model: Devices transmit their signals over a MAC.
+We first consider an AWGN channel, where the additive noise
+vector, denoted by z ∈ Cq, is assumed to be independent and
+identically distributed (i.i.d.) according to the complex normal
+distribution CN(0, σ2
+z). The received signal is given by
+y = x1 + x2 + z.
+(1)
+We also consider a slow fading MAC, where the fading
+coefficients, denoted by h1 and h2 ∈ C, are assumed to remain
+constant during each retrieval task, but changes across tasks in
+an i.i.d. fashion according to CN(0, σ2
+h). For the fading MAC,
+the received signal is given by:
+y = h1x1 + h2x2 + z.
+(2)
+The power allocation scheme and the end-to-end perfor-
+mance depend on the available CSI (SNR and channel gain)
+and the short-term power constraint imposed on each trans-
+mitter: 1
+q||xn||2
+2 ≤ 1
+i = 1, 2.
+Next, we will consider and compare three alternative trans-
+mission schemes, separation-based transmission, JSCC with
+OMA, and JSCC with NOMA, as well as the single-user
+benchmark from [4].
+A. Separate Digital Transmission
+In the digital scheme, transmitter Ei extracts a feature vector
+vi ∈ Rp from the source si, which is quantized to ˜vi ∈ Zp,
+and then mapped to a channel codeword xi ∈ Cq. The two
+transmitters transmit their codewords over the MAC channel.
+The receiver first decodes the two channel codewords to
+recover the quantized source signals ˜v1 and ˜v2. In the asymp-
+totic limit of infinite blocklength, the transmitted codewords
+can be decoded with a vanishing error probability if the
+transmission rates are within the capacity of the corresponding
+channels. In that case, the only source of error in the compu-
+tation of the desired function is the quantization. Note that
+the channel capacity provides only an upper bound on the
+maximum reliable communication rate, and is not achievable
+in practice, particularly at the very short blocklengths consid-
+ered here. The receiver then performs the retrieval task on the
+recovered source signals.
+B. JSCC
+In this scheme, feature vectors are directly mapped to the
+channel input signals. Transmitter i maps si to the codeword
+xi. We consider two JSCC schemes:
+JSCC with OMA: Each transmitter is allocated half the
+available channel bandwidth, i.e., q
+2 channel uses. The receiver
+first decodes the received signals from the two transmitters to
+recover estimates ˆv1 and ˆv2 of the source signals, and then
+performs the retrieval task on the recovered feature vectors.
+JSCC with NOMA: In this scheme, each transmitter
+occupies the full channel bandwidth of q, and the transmitted
+
+3
+codewords overlap. The receiver directly recovers estimates of
+the feature vectors from the received superposed signal, and
+performs the retrieval task.
+IV. DISTRIBUTED IMAGE RETRIEVAL
+In this section, we focus on the image retrieval task.
+A. Separate Digital Transmission
+Each transmitter consists of a feature encoder, modeled as
+a ResNet50 [20] network, followed by a feature compressor,
+employing quantization and arithmetic coding [4]. The com-
+pressed bits are then channel coded. The receiver decodes
+the received signal to obtain estimates of the feature vectors,
+which are then passed to the image retrieval module.
+Training strategy: We perform end-to-end training for the
+digital scheme, with the following loss function:
+l = 1
+3(lceaux1+lcemain+lceaux2)+λ·(log2 p(˜v1)+log2 p(˜v2)),
+(3)
+where lceaux1, lcemain, lceaux2 are the average cross-entropy
+losses between the ID predictions from three classifiers (two
+auxiliary classifiers and a main classifier, see Fig. 2), and the
+ground truth, and log2 p(˜v1) and log2 p(˜v2) are entropies of
+the feature vectors [4].
+B. JSCC
+In this scheme, the feature compressor, quantizer, arithmetic
+coder, and channel coder at the transmitter, and the channel
+decoder and arithmetic decoder at the receiver, are replaced
+by a single autoencoder architecture (see Fig. 2). The received
+signal is fed to two separate decoder modules, which decode
+estimates of the feature vectors sent by the two transmitters,
+as shown in Fig. 2. Once the feature vectors are recovered,
+they are used for the image retrieval task (see Fig. 2).
+Training strategy: A three-step training strategy is adopted,
+which consists of pre-training of the feature encoders (T1),
+pre-training of the JSCC autoencoders (T2), and end-to-end
+training (T3). In T1, the DNN feature encoder is pre-trained,
+using the average cross-entropy loss function:
+l = 1
+3(lceaux1 + lcemain + lceaux2).
+(4)
+In T2, the pre-trained feature encoders are frozen and only
+the JSCC autoencoders are trained, using the average mean
+squared error (MSE) loss between the transmitted and recon-
+structed feature vectors:
+l = 1
+2(lMSE1 + lMSE2).
+(5)
+In T3, the whole network is trained jointly, with the loss
+function in T1.
+We also propose a CSI-aware architecture variation for
+AWGN and slow fading channel with CSI at the receiver
+only (CSIR), where the available CSI (SNR or channel gain)
+is fed to the model via attention feature (AF) modules [9],
+[11] inserted before, after and between each layer of the
+autoencoder. For the AWGN channel, the AF modules at the
+Channel
+JSCC  
+decoder 1
+JSCC  
+decoder 2
+ 
+Feature  
+Encoder
+Feature 
+vector 
+JSCC  
+encoder 1
+Transmitter 1
+Feature  
+Encoder
+Feature 
+vector 
+JSCC  
+encoder 2
+Transmitter 2
+Receiver
+Main 
+classifier
+Auxiliary 
+classifier 2 
+Auxiliary 
+classifier 1 
+Auxiliary 
+ID predictions 1
+Main 
+ID predictions
+Auxiliary 
+ID predictions 2
+View-pooling  
+layer
+Image retrieval module
+ 
+Fig. 2: DNN architecture for the JSCC transmission schemes.
+encoder and decoder scale the intermediate feature maps to
+adapt to the channel SNR. For slow fading with CSIR, the AF
+modules scale the received signal and the intermediate feature
+maps by a channel-dependent constant, intuitively playing the
+role of channel equalisation.
+V. RESULTS
+A. Performance against channel SNR
+The proposed schemes for JSCC with OMA and NOMA are
+trained and tested on a pre-processed Market-1501 [21] dataset
+over a wide range of channel SNRs from -6dB to 15dB, and
+compared with the separation-based scheme and the single-
+device JSCC scheme in [4].
+In Fig. 3a, we plot the top-1 accuracy in an AWGN channel.
+In Fig. 3b, we plot the top-1 accuracy in a slow fading channel
+without CSI at the receiver. The digital scheme is not plotted
+in Fig. 3b because such a scheme is not possible to decode
+without CSI at the receiver, while JSCC allows communication
+even without the availability of CSI at the receiver. In Fig. 3c,
+we plot the top-1 accuracy in a slow fading channel with CSI
+available at the receiver. As expected, CSIR provides better
+accuracy than when CSI is absent at the receiver.
+In Figs. 3a, 3b and 3c, the proposed JSCC schemes out-
+perform the separate digital scheme at almost all SNRs,
+except at high SNRs. However, note that we assume MAC
+capacity-achieving codes with equal rate allocation for each
+transmitter in this separate digital scheme, and therefore the
+reported performance of the digital scheme is not achievable
+in practice, particularly for the very low channel bandwidth
+of q = 32 per user considered here. The two-device JSCC
+schemes outperform the single-device JSCC scheme for a
+wide range of channel SNRs, especially higher SNRs, showing
+that incorporating two views of the same identity to make a
+collaborative decision at the edge server improves the retrieval
+performance. It is also observed in Fig. 3a, 3b and 3c that
+JSCC with NOMA outperforms its orthogonal counterpart.
+In Fig. 3a, it is shown that while the OMA JSCC scheme
+outperforms the single-device JSCC benchmark at most SNRs,
+they are surpassed by it at very low SNRs. This is because, in
+the low SNR regime, it is more beneficial to allocate all the
+channel resources to one transmitter to acquire the features
+from that one with sufficient quality for retrieval, rather than
+
+4
+(a) AWGN channel
+(b) Slow fading channel without CSI
+(c) Slow fading channel with CSIR
+Fig. 3: Top-1 retrieval accuracies of the proposed two-device schemes and the single-device scheme under different channel
+SNRs, with a total channel bandwidth of q = 64.
+Scheme
+Squared cosine similarity
+OMA (AWGN)
+0.0151
+OMA (slow fading)
+0.0165
+NOMA (AWGN)
+0.7523
+NOMA (slow fading)
+0.8234
+TABLE I: Squared cosine similarity between input symbols of
+the OMA and NOMA schemes.
+receiving very low quality features from two queries. However,
+the NOMA JSCC scheme brings the benefits of both schemes
+together, and outperforms both schemes at all SNRs. In Fig.
+3c, the single-device JSCC as well as the proposed two-
+device JSCC schemes (both OMA and NOMA) outperform
+the separation-based scheme. These observations match our
+expectations. The suboptimality of separate source and channel
+coding used in the digital transmission scheme stems from two
+reasons. First of them is the usual suboptimality of separation
+in the finite blocklength regime. This was already observed in
+[4] for a point-to-point scenario. On the other hand, even in the
+infinite blocklength regime, separation becomes suboptimal
+when the two sources transmitted over the MAC are correlated.
+It is known that exploiting the correlation between the sources
+to generate correlated codewords at the encoders can strictly
+increase the end-to-end performance [22], [23]. To allow
+partial cooperation between the distributed transmitters, we
+must allow the transmitted signals to depend statistically on
+the source outputs, thus inducing correlation between the
+transmitted signals. Separation-based schemes operate in the
+opposite manner, where the dependence between the sources is
+destroyed by separate source and channel coding, thus making
+the transmitted signals independent.
+We observe that the orthogonal JSCC architecture learns
+to transmit uncorrelated signals, as shown in Table I, where
+the correlation between x1 and x2 is computed using squared
+cosine similarity, defined as cos2(x1, x2) ≜
+⟨x1,x2⟩2
+∥x1∥2∥x2∥2 . By
+sending independent symbols, the JSCC encoders capture non-
+overlapping information from the two views, thus avoiding
+redundancy, and maximising the use of communication re-
+sources. However, this mechanism is unable to make the
+distributed transmitters cooperate through the dependence of
+transmitted signals; hence, the lower accuracy achieved com-
+pared to the NOMA scheme. In contrast, JSCC with NOMA
+learns to transmit correlated signals. Higher correlation be-
+tween the transmitted signals for the NOMA scheme results
+in higher performance. In fact, in Fig. 4a, we plot the effect
+of the amount of correlation between the transmitted signals
+on the performance of the NOMA JSCC scheme, which we
+control by introducing a cosine similarity regularization term
+in the loss function as follows:
+l = 1
+3(lceaux1 + lcemain + lceaux2) + λ cos2 (x1, x2).
+(6)
+By using higher values of λ, we impose a higher penalty
+on the correlation between the transmitted signals, thus forcing
+x1 and x2 to be less correlated. We observe in Fig. 4a that
+the accuracy drops as the correlation between the transmit-
+ted signals decreases. Interestingly, if the cosine similarity
+between the symbols transmitted by the two transmitters in
+the NOMA scheme is reduced using the regularisation term,
+its accuracy approaches that of the orthogonal JSCC scheme
+when the cosine similarity approaches 0.
+B. SNR-aware JSCC
+The SNR-aware JSCC scheme, introduced in Section IV-B,
+is trained over a range of SNRtrain, and tested over a wide
+range of SNRtest values, from -6 to 18dB. In Fig. 4b and
+Fig. 4c, the performance of the SNR-aware schemes for the
+two JSCC schemes is compared with that of non-SNR-aware
+architectures trained over a single SNRtrain but tested on
+different SNRtest values - unlike in Fig. 3, where we have
+matching training and test SNR values.
+Note that the non-SNR-aware architectures exhibit graceful
+degradation when there is channel mismatch, that is, when
+the test channel conditions are worse than that of the training
+conditions. Thus, the JSCC scheme is able to avoid the cliff
+effect which conventional digital communication suffers from,
+where the performance of the digital schemes drops sharply
+when channel conditions are worse than those for which the
+encoder and decoder are designed. However, the SNR-aware
+architectures are observed to achieve strictly higher retrieval
+
+O.B 
+0.7
+0.6
+Accuracy
+50
+p-1
+0.4
+0.3
+Two-source NOMA-ISCC
+Two-source OMA-jSCC
+0.2
+Single-source JSCC
+Two-source Digital scheme
+5
+5
+15
+Channel SNR (dB)0.7
+0.6
+op-1 Accuracy
+50
+t0
+E'0
+Two-source NOMA-JSCC
+0.2
+Two-source OMA-JScC
+Single-source JScC
+5
+0
+5
+1
+15
+Channel SNR (dB)0.B
+200
+0.7
+18D
+0.6
+160
+Top-1 Accuracy
+50
+140
+0.4
+120
+0.3
+130
+Two-source NOMA-JSCC
+0.2
+Two-source OMA-JSCC
+Single-source jScC
+Two-source Digital scheme
+0.1
+5
+0
+5
+15
+Channel SNR (dB)5
+(a) JSCC with NOMA - cosine similarity
+(b) JSCC with OMA - AWGN channel
+(c) JSCC with NOMA - AWGN channel
+Fig. 4: Top-1 retrieval accuracies of: (a) the JSCC-NOMA scheme on AWGN and slow fading channels against different
+squared cosine similarity between x1 and x2, with channel SNR = 0dB, and channel bandwidth q = 64, and (b),(c) the
+SNR-aware scheme and the original schemes trained with various SNRtrain values against different SNRtest values for OMA
+and NOMA schemes.
+accuracies than the non-SNR-aware architectures (see Fig. 4b
+and 4c), providing a single DNN that performs the same or
+better on all SNRs than employing a distinct DNN optimised
+for each particular SNR value or range.
+VI. CONCLUSION
+We proposed two JSCC schemes for deep-learning based
+distributed retrieval at the wireless edge, with OMA and
+NOMA, respectively. These schemes are shown to outper-
+form conventional separation based alternative with capacity-
+achieving channel codes, and the JSCC scheme with a single
+source [4]. We observed that the NOMA JSCC scheme out-
+performs its OMA counterpart with TDMA. We also observed
+that the DNN architecture, when trained for NOMA, learns
+to transmit correlated signals to induce partial cooperation
+between the transmitters and to improve the final accuracy.
+The OMA JSCC scheme, in contrast, learns to transmit un-
+correlated signals. With these observations in mind, in our
+future work, we will study how the correlation between the
+transmitted signals can be optimized to improve performance.
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+
+0.725
+0.675
+Accurad
+0.65
+p-1
+0.625
+0.600
+0.575
+AWGN
+0.550
+Slow fading without CSl
+0.1
+0.2
+0.3
+0.5
+0.6
+0.7
+0.B
+Squared cosine similarity0.B
+0.7
+0.6
+Accurat
+Top-1 
+0.5
+SNR-aware model
+t0
+SNR train= 12dB
+SNR train = 6dB
+SNR train = OdB
+0.3
+SNR train = -6dB
+5
+5
+15
+SWRtest (dB)0.B0
+0.75
+0.70
+0.65
+0.60
+SNR-aware model
+0.55
+SNRtrain=12dB
+SNR train = 6dB
+SNR train = OdB
+0.50
+SNR train = -6dB
+5
+0
+15
+SWRtest (dB)

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@@ -0,0 +1,425 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf,len=424
+page_content='1  Collaborative Semantic Communication at the Edge  Wing Fei Lo, Nitish Mital, Member, IEEE, Haotian Wu, Graduate Student Member, IEEE,  Deniz G¨und¨uz, Fellow, IEEE  Abstract—We study the collaborative image retrieval problem  at the wireless edge, where multiple edge devices capture images  of the same object from different angles and locations, which  are then used jointly to retrieve similar images at the edge  server over a shared multiple access channel (MAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We propose  two novel deep learning-based joint source and channel coding  (JSCC) schemes for the task over both additive white Gaussian  noise (AWGN) and Rayleigh slow fading channels, with the aim  of maximizing the retrieval accuracy under a total bandwidth  constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The proposed schemes are evaluated on a wide  range of channel signal-to-noise ratios (SNRs), and shown to  outperform the single-device JSCC and the separation-based  multiple-access benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We also propose two novel SNR-  aware JSCC schemes with attention modules to improve the  performance in the case of channel mismatch between training  and test instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Index Terms—Semantic communication, Internet of Things,  person re-identification, deep joint source and channel coding,  collaborative image retrieval  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' INTRODUCTION  I  N recent years, machine learning tasks at the wireless edge  have been studied extensively in the literature, including  distributed inference problems over wireless channels [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  In distributed inference problems, it is often assumed that  centrally trained models, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' deep neural networks (DNNs)  are employed across multiple distributed nodes, which have  limited communication resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Communication is essential  in scenarios in which data is available at different nodes, and  exploiting this data can increase the inference accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In  particular, in image retrieval, image of an object or a person  taken by an edge device is used to identify the images of the  same object or person in a gallery database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The images in the  database may be taken by different cameras, from different  angles, and at different times, making re-identification (ReID)  a highly challenging inference problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Note that, unlike most  conventional classification or regression problems, which can  be carried out locally at the edge device, for image retrieval,  remote inference is essential even if the edge devices have  unlimited computational power, as the gallery database is only  available at the edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' On the other hand, due to latency  and bandwidth constraints, sending the whole image via the  wireless channel is not feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Instead, learning-based feature  extraction is done at the edge, and only the most important  features of the source image should be sent to the edge server  through the wireless channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  In [4], both separation-based and joint source channel  coding (JSCC) approaches have been studied for feature trans-  mission in remote image retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' While Shannon’s separation  The authors are with the Department of Electrical and Electronic Engi-  neering, Imperial College London, London SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' (e-mail: hao-  tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='wu17@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='uk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 1: Illustration of the two-device collaborative image  retrieval problem at the wireless edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  theorem [5] states that separating source and channel coding  can achieve asymptotic optimality, this theorem breaks down  in finite block-lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We typically have much more stringent  latency constraints on edge inference applications compared  to the delivery of images or videos;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' hence, our interest is in  the very short blocklengths, where the separation typically has  very poor performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' An autoencoder-based JSCC (JSCC-  AE) scheme is proposed in [4], and it is shown to outperform  its digital counterpart under all channel conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  In this paper, we study the collaborative ReID problem,  where two edge devices capture images of the same object,  which are then used to identify similar images in a gallery  database at an edge server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The increasing number of edge  devices raises new requirements for collaborative inference:  edge devices must collaborate not only with the edge server,  but also with each other as multiple images can provide addi-  tional information and can potentially improve ReID accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  The goal of this paper is to develop a deep learning-based  JSCC scheme for the two-device scenario, which maximizes  the accuracy of the retrieval task while communicating over a  shared multiple access channel (MAC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We first consider an  orthogonal multiple access (OMA) scheme employing time  division multiple access (TDMA) with distributed JSCC, and  show that it outperforms the schemes in [4], as well as a  conventional separate source-channel coding scheme, where  each device transmits a quantized version of its features  to the receiver using capacity-achieving channel codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In  addition, we study an alternative non-orthogonal multiple  access (NOMA) approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Benefits of NOMA transmission  in various distributed inference and training problems have  recently received significant interest [6]–[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In this approach,  our goal is exploit the superposition property of the wireless  medium, and the features transmitted as analog values over the  shared wireless channel get aggregated “over-the-air” rather  than interfering with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We evaluate these schemes on  the additive white Gaussian noise (AWGN) and Rayleigh slow  fading channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Inspired by the attention mechanism in adap-  tive JSCC [9]–[11], we also propose an SNR-aware scheme  for the AWGN channel to adjust the networks depending on  the SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='03996v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='IV]  10 Jan 2023  ()  Pre-processing  Edge device 1  (α)  Wireless  ID  channel  prediction  ()  Pedestrian  Pre-processing  Edge server  Edge device 22  Our main contributions can be summarized as follows:  To the best of our knowledge, this is the first paper  to study collaborative inference among edge devices for  joint retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We propose two new collaborative JSCC  schemes for OMA and NOMA transmissions, and show  the superiority of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  We construct and analyze DNN architectures for a chan-  nel state information (CSI)-aware JSCC scheme (SNR-  aware and channel fading-aware), where a single network  is trained to exploit the channel state information for  channel equalization and SNR-adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' RELATED WORK  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Image retrieval  Image retrieval task aims to improve the quality of identity  recognition, particularly in surveillance applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Given a  query image, an image retrieval model assesses its similarities  with gallery images, and matches is to the ‘nearest’ ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Performance can be evaluated through top-1 retrieval accuracy  [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Image retrieval task has received significant attention in  recent years [13] thanks to the tremendous success of deep  learning technologies [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Remote inference at the wireless edge  Classical communication systems are designed to deliver  source signals, such as images, audio, or video, to a re-  ceiver with the highest end-to-end fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' However, with  the rapid growth of machine intelligence and the associated  machine-to-machine communications, the goal of emergent  communication systems is shifting towards making accurate  inferences about a remote signal rather than reconstructing  it [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Literature on joint edge-device inference [15], [16]  mostly focus on rate-limited scenario, and ignore the channel  effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Jankowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' proposed a retrieval-oriented image  compression scheme and a JSCC scheme for the retrieval  task [4] with state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Remote inference  problems are also attracting significant interest in the context  of the emerging semantic communication paradigm [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Multi-device collaborative learning  Existing multi-device algorithms mainly focus on image  transmission [18] and classification tasks [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Shao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  proposed a task-oriented communication scheme for multi-  device collaborative edge inference [19], which utilizes the  information bottleneck (IB) principle for feature extraction and  the deterministic distributed information bottleneck (DDIB)  principle for distributed feature encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Different from pre-  vious work, our paper explores cooperation for the image  retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' SYSTEM MODEL  We consider two transmitters, each of them having access  to images of the same object taken by a different camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  We denote the image observed by transmitter i by si ∈ Rp,  i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Transmitter i employs an encoding function Ei :  Rp → Cq, where xi = Ei(si) ∈ Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Here, q represents the  available channel bandwidth, and r ≜ q  p is the bandwidth ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  The decoder function D : Cq → D is employed at the receiver,  where D ≡ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' , D}, and D is the size of the database,  maps the received signal y to the result of the retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Channel model: Devices transmit their signals over a MAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  We first consider an AWGN channel, where the additive noise  vector, denoted by z ∈ Cq, is assumed to be independent and  identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=') according to the complex normal  distribution CN(0, σ2  z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The received signal is given by  y = x1 + x2 + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  (1)  We also consider a slow fading MAC, where the fading  coefficients, denoted by h1 and h2 ∈ C, are assumed to remain  constant during each retrieval task, but changes across tasks in  an i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' fashion according to CN(0, σ2  h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' For the fading MAC,  the received signal is given by:  y = h1x1 + h2x2 + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  (2)  The power allocation scheme and the end-to-end perfor-  mance depend on the available CSI (SNR and channel gain)  and the short-term power constraint imposed on each trans-  mitter: 1  q||xn||2  2 ≤ 1  i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Next, we will consider and compare three alternative trans-  mission schemes, separation-based transmission, JSCC with  OMA, and JSCC with NOMA, as well as the single-user  benchmark from [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Separate Digital Transmission  In the digital scheme, transmitter Ei extracts a feature vector  vi ∈ Rp from the source si, which is quantized to ˜vi ∈ Zp,  and then mapped to a channel codeword xi ∈ Cq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The two  transmitters transmit their codewords over the MAC channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  The receiver first decodes the two channel codewords to  recover the quantized source signals ˜v1 and ˜v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In the asymp-  totic limit of infinite blocklength, the transmitted codewords  can be decoded with a vanishing error probability if the  transmission rates are within the capacity of the corresponding  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In that case, the only source of error in the compu-  tation of the desired function is the quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Note that  the channel capacity provides only an upper bound on the  maximum reliable communication rate, and is not achievable  in practice, particularly at the very short blocklengths consid-  ered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The receiver then performs the retrieval task on the  recovered source signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' JSCC  In this scheme, feature vectors are directly mapped to the  channel input signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Transmitter i maps si to the codeword  xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We consider two JSCC schemes:  JSCC with OMA: Each transmitter is allocated half the  available channel bandwidth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=', q  2 channel uses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The receiver  first decodes the received signals from the two transmitters to  recover estimates ˆv1 and ˆv2 of the source signals, and then  performs the retrieval task on the recovered feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  JSCC with NOMA: In this scheme, each transmitter  occupies the full channel bandwidth of q, and the transmitted  3  codewords overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The receiver directly recovers estimates of  the feature vectors from the received superposed signal, and  performs the retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' DISTRIBUTED IMAGE RETRIEVAL  In this section, we focus on the image retrieval task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Separate Digital Transmission  Each transmitter consists of a feature encoder, modeled as  a ResNet50 [20] network, followed by a feature compressor,  employing quantization and arithmetic coding [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The com-  pressed bits are then channel coded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The receiver decodes  the received signal to obtain estimates of the feature vectors,  which are then passed to the image retrieval module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Training strategy: We perform end-to-end training for the  digital scheme, with the following loss function:  l = 1  3(lceaux1+lcemain+lceaux2)+λ·(log2 p(˜v1)+log2 p(˜v2)),  (3)  where lceaux1, lcemain, lceaux2 are the average cross-entropy  losses between the ID predictions from three classifiers (two  auxiliary classifiers and a main classifier, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 2), and the  ground truth, and log2 p(˜v1) and log2 p(˜v2) are entropies of  the feature vectors [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' JSCC  In this scheme, the feature compressor, quantizer, arithmetic  coder, and channel coder at the transmitter, and the channel  decoder and arithmetic decoder at the receiver, are replaced  by a single autoencoder architecture (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The received  signal is fed to two separate decoder modules, which decode  estimates of the feature vectors sent by the two transmitters,  as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Once the feature vectors are recovered,  they are used for the image retrieval task (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Training strategy: A three-step training strategy is adopted,  which consists of pre-training of the feature encoders (T1),  pre-training of the JSCC autoencoders (T2), and end-to-end  training (T3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In T1, the DNN feature encoder is pre-trained,  using the average cross-entropy loss function:  l = 1  3(lceaux1 + lcemain + lceaux2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  (4)  In T2, the pre-trained feature encoders are frozen and only  the JSCC autoencoders are trained, using the average mean  squared error (MSE) loss between the transmitted and recon-  structed feature vectors:  l = 1  2(lMSE1 + lMSE2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  (5)  In T3, the whole network is trained jointly, with the loss  function in T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  We also propose a CSI-aware architecture variation for  AWGN and slow fading channel with CSI at the receiver  only (CSIR), where the available CSI (SNR or channel gain)  is fed to the model via attention feature (AF) modules [9],  [11] inserted before, after and between each layer of the  autoencoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' For the AWGN channel, the AF modules at the  Channel  JSCC  decoder 1  JSCC  decoder 2  Feature  Encoder  Feature  vector  JSCC  encoder 1  Transmitter 1  Feature  Encoder  Feature  vector  JSCC  encoder 2  Transmitter 2  Receiver  Main  classifier  Auxiliary  classifier 2  Auxiliary  classifier 1  Auxiliary  ID predictions 1  Main  ID predictions  Auxiliary  ID predictions 2  View-pooling  layer  Image retrieval module  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 2: DNN architecture for the JSCC transmission schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  encoder and decoder scale the intermediate feature maps to  adapt to the channel SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' For slow fading with CSIR, the AF  modules scale the received signal and the intermediate feature  maps by a channel-dependent constant, intuitively playing the  role of channel equalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' RESULTS  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Performance against channel SNR  The proposed schemes for JSCC with OMA and NOMA are  trained and tested on a pre-processed Market-1501 [21] dataset  over a wide range of channel SNRs from -6dB to 15dB, and  compared with the separation-based scheme and the single-  device JSCC scheme in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3a, we plot the top-1 accuracy in an AWGN channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3b, we plot the top-1 accuracy in a slow fading channel  without CSI at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The digital scheme is not plotted  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3b because such a scheme is not possible to decode  without CSI at the receiver, while JSCC allows communication  even without the availability of CSI at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3c,  we plot the top-1 accuracy in a slow fading channel with CSI  available at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' As expected, CSIR provides better  accuracy than when CSI is absent at the receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  In Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3a, 3b and 3c, the proposed JSCC schemes out-  perform the separate digital scheme at almost all SNRs,  except at high SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' However, note that we assume MAC  capacity-achieving codes with equal rate allocation for each  transmitter in this separate digital scheme, and therefore the  reported performance of the digital scheme is not achievable  in practice, particularly for the very low channel bandwidth  of q = 32 per user considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The two-device JSCC  schemes outperform the single-device JSCC scheme for a  wide range of channel SNRs, especially higher SNRs, showing  that incorporating two views of the same identity to make a  collaborative decision at the edge server improves the retrieval  performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' It is also observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3a, 3b and 3c that  JSCC with NOMA outperforms its orthogonal counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3a, it is shown that while the OMA JSCC scheme  outperforms the single-device JSCC benchmark at most SNRs,  they are surpassed by it at very low SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' This is because, in  the low SNR regime, it is more beneficial to allocate all the  channel resources to one transmitter to acquire the features  from that one with sufficient quality for retrieval, rather than  4  (a) AWGN channel  (b) Slow fading channel without CSI  (c) Slow fading channel with CSIR  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3: Top-1 retrieval accuracies of the proposed two-device schemes and the single-device scheme under different channel  SNRs, with a total channel bandwidth of q = 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Scheme  Squared cosine similarity  OMA (AWGN)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='0151  OMA (slow fading)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='0165  NOMA (AWGN)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='7523  NOMA (slow fading)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='8234  TABLE I: Squared cosine similarity between input symbols of  the OMA and NOMA schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  receiving very low quality features from two queries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' However,  the NOMA JSCC scheme brings the benefits of both schemes  together, and outperforms both schemes at all SNRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  3c, the single-device JSCC as well as the proposed two-  device JSCC schemes (both OMA and NOMA) outperform  the separation-based scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' These observations match our  expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' The suboptimality of separate source and channel  coding used in the digital transmission scheme stems from two  reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' First of them is the usual suboptimality of separation  in the finite blocklength regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' This was already observed in  [4] for a point-to-point scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' On the other hand, even in the  infinite blocklength regime, separation becomes suboptimal  when the two sources transmitted over the MAC are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  It is known that exploiting the correlation between the sources  to generate correlated codewords at the encoders can strictly  increase the end-to-end performance [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' To allow  partial cooperation between the distributed transmitters, we  must allow the transmitted signals to depend statistically on  the source outputs, thus inducing correlation between the  transmitted signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Separation-based schemes operate in the  opposite manner, where the dependence between the sources is  destroyed by separate source and channel coding, thus making  the transmitted signals independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  We observe that the orthogonal JSCC architecture learns  to transmit uncorrelated signals, as shown in Table I, where  the correlation between x1 and x2 is computed using squared  cosine similarity, defined as cos2(x1, x2) ≜  ⟨x1,x2⟩2  ∥x1∥2∥x2∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' By  sending independent symbols, the JSCC encoders capture non-  overlapping information from the two views, thus avoiding  redundancy, and maximising the use of communication re-  sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' However, this mechanism is unable to make the  distributed transmitters cooperate through the dependence of  transmitted signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' hence, the lower accuracy achieved com-  pared to the NOMA scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In contrast, JSCC with NOMA  learns to transmit correlated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Higher correlation be-  tween the transmitted signals for the NOMA scheme results  in higher performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In fact, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 4a, we plot the effect  of the amount of correlation between the transmitted signals  on the performance of the NOMA JSCC scheme, which we  control by introducing a cosine similarity regularization term  in the loss function as follows:  l = 1  3(lceaux1 + lcemain + lceaux2) + λ cos2 (x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  (6)  By using higher values of λ, we impose a higher penalty  on the correlation between the transmitted signals, thus forcing  x1 and x2 to be less correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 4a that  the accuracy drops as the correlation between the transmit-  ted signals decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Interestingly, if the cosine similarity  between the symbols transmitted by the two transmitters in  the NOMA scheme is reduced using the regularisation term,  its accuracy approaches that of the orthogonal JSCC scheme  when the cosine similarity approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' SNR-aware JSCC  The SNR-aware JSCC scheme, introduced in Section IV-B,  is trained over a range of SNRtrain, and tested over a wide  range of SNRtest values, from -6 to 18dB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 4b and  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 4c, the performance of the SNR-aware schemes for the  two JSCC schemes is compared with that of non-SNR-aware  architectures trained over a single SNRtrain but tested on  different SNRtest values - unlike in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 3, where we have  matching training and test SNR values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  Note that the non-SNR-aware architectures exhibit graceful  degradation when there is channel mismatch, that is, when  the test channel conditions are worse than that of the training  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Thus, the JSCC scheme is able to avoid the cliff  effect which conventional digital communication suffers from,  where the performance of the digital schemes drops sharply  when channel conditions are worse than those for which the  encoder and decoder are designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' However, the SNR-aware  architectures are observed to achieve strictly higher retrieval  O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='B  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='6  Accuracy  50  p-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='3  Two-source NOMA-ISCC  Two-source OMA-jSCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='2  Single-source JSCC  Two-source Digital scheme  5  5  15  Channel SNR (dB)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content="6  op-1 Accuracy  50  t0  E'0  Two-source NOMA-JSCC  0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='2  Two-source OMA-JScC  Single-source JScC  5  0  5  1  15  Channel SNR (dB)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='B  200  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='7  18D  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='6  160  Top-1 Accuracy  50  140  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='4  120  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='3  130  Two-source NOMA-JSCC  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='2  Two-source OMA-JSCC  Single-source jScC  Two-source Digital scheme  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='1  5  0  5  15  Channel SNR (dB)5  (a) JSCC with NOMA - cosine similarity  (b) JSCC with OMA - AWGN channel  (c) JSCC with NOMA - AWGN channel  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 4: Top-1 retrieval accuracies of: (a) the JSCC-NOMA scheme on AWGN and slow fading channels against different  squared cosine similarity between x1 and x2, with channel SNR = 0dB, and channel bandwidth q = 64, and (b),(c) the  SNR-aware scheme and the original schemes trained with various SNRtrain values against different SNRtest values for OMA  and NOMA schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  accuracies than the non-SNR-aware architectures (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' 4b  and 4c), providing a single DNN that performs the same or  better on all SNRs than employing a distinct DNN optimised  for each particular SNR value or range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' CONCLUSION  We proposed two JSCC schemes for deep-learning based  distributed retrieval at the wireless edge, with OMA and  NOMA, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' These schemes are shown to outper-  form conventional separation based alternative with capacity-  achieving channel codes, and the JSCC scheme with a single  source [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We observed that the NOMA JSCC scheme out-  performs its OMA counterpart with TDMA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' We also observed  that the DNN architecture, when trained for NOMA, learns  to transmit correlated signals to induce partial cooperation  between the transmitters and to improve the final accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  The OMA JSCC scheme, in contrast, learns to transmit un-  correlated signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' With these observations in mind, in our  future work, we will study how the correlation between the  transmitted signals can be optimized to improve performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='  REFERENCES  [1] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' G¨und¨uz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Kurka, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' Jankowski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
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+page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
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+page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
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+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
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+page_content='675  Accurad  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='65  p-1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
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+page_content='575  AWGN  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
+page_content='550  Slow fading without CSl  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
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+page_content='55  SNRtrain=12dB  SNR train = 6dB  SNR train = OdB  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dE2T4oBgHgl3EQfmQc9/content/2301.03996v1.pdf'}
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@@ -0,0 +1,367 @@
+The Gauge issue and the Hamiltonian theory of cosmological perturbations
+Alice Boldrin1, ∗
+1National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa, Poland
+We present a general formalism for the Hamiltonian description of perturbation theory
+around any spatially homogeneous spacetime. We employ and refine the Dirac method for
+constrained systems, which is very well-suited to cosmological perturbations. This approach
+includes a discussion of the gauge-invariant dynamics of perturbations as well as an analysis
+of gauge transformations, gauge-fixing, partial gauge-fixing and spacetime reconstruction.
+We will introduce the Kuchaˇr parametrization of the kinematical phase space as a conve-
+nient tool for studying the gauge transformations. The key element of this approach is the
+reconstruction of spacetime based on gauge-fixing conditions.
+I.
+INTRODUCTION
+In the attempt to obtain a quantum theory suitable for the description of the primordial struc-
+ture of the Universe, we study the Hamiltonian formalism for cosmological perturbation theory
+(CPT). This work has been done before with different background spacetime models like the Fried-
+man universe [1] and the Bianchi Type I model [2]. Our aim is to study the complete Hamiltonian
+formalism in a general background focusing on the gauge independent description of CPT as well
+as the issue of gauge fixing (see e.g. [3, 4] for alternative discussions on the gauge issue in CPT),
+gauge transformations and spacetime reconstruction. We employ the Dirac method [5] to study the
+Hamiltonian in different gauges and reconstruct the spacetime metric from gauge-invariant quan-
+tities (Dirac observables). We also discuss an alternative method based on the so-called Kuchaˇr
+decomposition [6] which provides a parametrization of the phase space in which the constrains
+play the role of canonical variables conjugate to the gauge-fixing conditions. For a more detailed
+discussion and for an application of the presented method, see [7].
+∗ [email protected]
+arXiv:2301.08646v1  [gr-qc]  20 Jan 2023
+
+2
+II.
+COSMOLOGICAL PERTURBATION THEORY
+The Hamiltonian in the the Arnowitt-Deser-Misner (ADM) formalism [8] expanded to second
+order reads1
+H =
+�
+T3
+�
+NH(0)
+0
++ NH(2)
+0
++ δN µδHµ
+�
+d3x
+(1)
+where N is the background lapse function and δN µ, with µ = 0, i, are the first order lapse and
+shift functions. The Hamiltonian densities H(0) and H(2) are respectively zeroth and second order,
+whereas δHµ represent the first order constraints. We assume a spatially homogeneous background
+spacetime with spatial coordinates defined such that the background shift vector Ni vanishes as
+well as the background Hamiltonian H(0)
+i . The Hamiltonian (1) is a function of the background
+canonical variables ¯qij and ¯πij which are respectively the three-metric and three-momenta, and the
+perturbed variables defined as δqij = qij − qij and δπij = πij − πij.
+The Hamiltonian (1) defines a gauge for the following reasons:
+First, at each spatial point the constraints algebra is closed, i.e.
+{δHi, δHj} = 0,
+{δH0, δHi} = 0,
+(2)
+where this result is true for any homogeneous background. Furthermore the constraints are dy-
+namically stables, i.e.
+{H, δH0} = −δHi
+,i(x) ≈ 0,
+{H, δHi} = 0,
+(3)
+where the ”weak equality” ≈, means that the equality holds in the constraint surface.
+III.
+GAUGE-FIXING AND DIRAC PROCEDURE
+The four constraints δHµ generate a gauge freedom which can be removed by imposing four
+gauge-fixing conditions δcµ = 0. The Poisson bracket between the gauge-fixing conditions and the
+constraints form an invertible matrix det{δcµ, δHµ} ̸= 0. Applying the constraints and the gauge-
+fixing conditions we can reduce our Hamiltonian which will now depend on 4 physical variables
+(δqphys
+I
+, δπI
+phys) instead of the 12 ADM perturbation2 variables (δqij, δπij). Those new variables
+form a canonical coordinate system on the submanifold in the kinematical phase space.
+This
+1 We assume the topology of the spacetime to be M ≃ T3 × R so to have a spatially compact universe and avoid
+ambiguous definitions of the symplectic structure for background (homogeneous) variables.
+2 We assume the vacuum case for the sake of clarity. See [1] or [2] for the Dirac method applied when there is matter
+content.
+
+3
+submanifold is thus called the physical phase space3.
+The parametrization provided by these
+physical variables is defined by the gauge-fixing surface that intersects all gauge orbits (see Fig. 1
+). It is convenient to define a set of gauge-independent variables defined as
+{δDI, δHµ} ≈ 0, ∀µ,
+(4)
+which parametrize the space of gauge orbits in the constraints surface. Those variables are known
+as Dirac observables and are equal to the number of physical variables. There exists a one-to-one
+correspondence between the Dirac observables and the physical variables, such that
+δDI + ϵµ
+I δcµ + ξµ
+I δHµ = δOphys
+I
+(δqphys
+I
+, δπI
+phys)
+(5)
+where ϵµ
+I and ξµ
+I are background coefficients. Using this new parametrization the Hamiltonian can
+be written in a gauge-independent manner as H(2)
+phys = H(2)
+red + H(2)
+ext, where H(2)
+phys denotes the so
+called physical Hamiltonian, H(2)
+red is the reduced Hamiltonian in terms of the physical variables and
+H(2)
+ext is the extra Hamiltonian generated by the time-dependent canonical transformation needed
+to change parametrization.
+FIG. 1. Graphical representation of the Dirac procedure.
+IV.
+SPACETIME RECONSTRUCTION
+In the previous section we discussed how to obtain the physical Hamiltonian.
+In order to
+reconstruct the spacetime we still need to find the values of the first-order lapse and shift. To do
+so we use the consistency equation {δcµ, H} = 0, which, from Eq. (1), implies
+δN µ
+N
+= −{δcν, δHµ}−1 �
+{δcν, δH(0)} + {δcν, H(2)}
+�
+(6)
+3 It’s canonical structure is now given by the Dirac brackets {., .}D = {., .} − {., δφµ}{δφµ, δφν}−1{δφν, .}, where
+δφµ ∈ (δHµ, δcµ).
+
+4
+This equation is only meaningful in the constraint surface.
+V.
+KUCHAˇR DECOMPOSITION
+We present a different parametrization of the kinematical phase space where the constraints
+take the role of canonical variables. For instance, we define two sets of canonical variables. The
+first set comprises the first order constraints δHµ and the 4 gauge-fixing functions, here denoted
+as δCµ. The second pair of canonical variables is given by the Dirac observables δDI, defined in
+Eq.(4). The Hamiltonian written in this parametrization will then be
+H → HK = H + K =
+� �
+NH(0)
+0
++ N(H(2)
+0
++ K) + δN µδHµ
+�
+d3x,
+(7)
+where K is the extra Hamiltonian coming from the time-dependent parametrization. We notice
+that, since the constraints are conserved in the constraint surface, terms of the form ∝ δCµδCν,
+∝ δQIδCµ and ∝ δPIδCµ are not present in Eq. (7). Moreover, considering that H(2) ≈ H(2)
+red
+and K(2) ≈ H(2)
+ext, which tells us the two dynamics must be weakly equal, we have that the total
+Hamiltonian can only be of the form
+HK =N
+� �
+H(2)
+phys(δQI, δP I)
+�
+��
+�
+physical part
++
++
+�
+λµI
+1 δQI + λµ
+2IδPI + λµν
+3 δHν + λµ
+4νδCν + δN µ
+N
+�
+Hµ
+�
+��
+�
+weakly vanishing part
+�
+d3x,
+(8)
+where λµI
+1 , λµ
+I2 and λµν
+3
+are zeroth-order coefficients that can depend on the gauge-fixing δCµ.
+The value of λµ
+4ν is gauge-invariant, it is showed to be fixed unambiguously by the algebra of the
+hypersurface (3).
+A.
+Gauge transformations
+An interesting property of the Kuchaˇr decomposition comes from the freedom in the choice of
+the canonical variable δCµ. This means that we have a class of parametrizations of the kinematical
+phase space. In particular, we can define the new set of gauge-fixing conditions as δ ˜Cµ, and the full
+gauge transformation will be given by the map G : (δHµ, δCµ, δQI, δP I) → (δ ˜Hµ, δ ˜Cµ, δ ˜QI, δ ˜P I),
+where δHµ = δ ˜Hµ. We are free to assume that the new gauge-fixing functions are thus canonically
+conjugate to the constraints δHµ. Thus we have {δHν, δ ˜Cµ − δCµ} = 0, which implies
+δ ˜Cµ = δCµ + αµ
+I δP I + βµIδQI + γµνδHν,
+(9)
+
+5
+where αµ
+I , βµI and γµν are background parameters.
+The gauge-fixing condition is only relevant in the constraints surface, so Eq.(9) is fully deter-
+mined by the parameters αµ
+I and βµI. Moreover it means that the space of gauge-fixing conditions
+is the affine space of dimension equal to the number od Dirac observables. The introduction of a
+different gauge will lead to the new Hamiltonian H ˜
+K, with an extra Hamiltonian density ∆K(2).
+Studying the new symplectic form of the system we find that γµν depends only on αµ
+I and βµI,
+which thus are the only parameters needed to uniquely determine the gauge transformation.
+B.
+Spacetime reconstruction
+As discussed in Sec. IV, the spacetime reconstruction is obtained by the dynamical equations
+δ ˙Cµ = 0, which means that it is sensitive to the chosen parametrization. In particular in the
+Kuchaˇr parametrization we will have {δCν, HK}K = 0, which, from Eq. (7) becomes
+δN µ
+N
+= −∂(H(2) + K(2))
+∂δHµ
+.
+(10)
+Notice the above formula only depends on the weakly vanishing part of the Hamiltonian since the
+lapse and shift are gauge-dependent quantities. It is interesting to consider the difference between
+the lapse and shift in two gauges. Using Eq. (8), we have
+δ ˜Nµ
+N
+����
+δ ˜Cµ=0
+− δN µ
+N
+����
+δCµ=0
+≈
+≈
+�
+�λµ
+4νβνI + ˙βµI +
+∂2H(2)
+phys
+∂δQI∂δP J βµJ −
+∂2H(2)
+phys
+∂δQI∂δQJ
+αµ
+J
+�
+� δQI
++
+�
+�λµ
+4ναν
+I + ˙αµ
+I −
+∂2H(2)
+phys
+∂δP I∂δQJ
+αµ
+J +
+∂2H(2)
+phys
+∂δP I∂δP J βµJ
+�
+� δP I.
+(11)
+We see that the spacetime reconstruction in a new gauge can be obtained by the lapse and shift in
+the initial gauge plus some terms which solely depend on the physical part of the Hamiltonian H(2)
+phys
+and the gauge-invariant coefficient λµ
+4ν, which can be obtained from the algebra of the hypersurface
+deformations.
+VI.
+PARTIAL GAUGE-FIXING
+We previously discussed the gauge-fixing defined as setting the conditions δCµ = 0. However it
+can be interesting to study the case in which these 4 conditions are substituted with conditions on
+
+6
+the lapse and shift functions. This is what we call partial gauge-fixing. From this consideration we
+can study the transformations which preserve the lapse and shift functions, that is, δ ˜
+Nµ
+N
+��
+δ ˜Cµ=0 −
+δNµ
+N
+��
+δCµ=0 = 0. Using Eq. (11) and solving it for αν
+I and βµI, we can solve the ambiguity in the
+choice of the gauge-fixing condition.
+˙αµ
+I = −βµJ
+∂2H(2)
+phys
+∂δP J∂δP I + αµ
+J
+∂2H(2)
+phys
+∂δQJ∂δP I − λµ
+4ναν
+I,
+˙βµI = −βµJ
+∂2H(2)
+phys
+∂δP J∂δQI
++ αµ
+J
+∂2H(2)
+phys
+∂δQJ∂δQI
+− λµ
+4νβνI,
+(12)
+The above equations fix the gauge-fixing function at all times once δCµ(t0) is fixed at an initial
+time t0. This means that the choice of δCµ(t0) fixes the initial three-surface. Given the initial
+values of the Dirac observables (δQI(t0), δP I(t0)), we are able to explicitly reconstruct the initial
+three-surface in terms of the ADM perturbation variables. Moreover we are able to fully reconstruct
+the spacetime geometry since the evolution of the three-surface with its coordinates is completely
+determined by the evolution of the gauge-fixing function δ ˜Cµ(t) and the independent evolution of
+the gauge-invariant variables4 (δQI(t), δP(t)).
+VII.
+CONCLUSIONS
+We were able to simplify the Hamiltonian approach to CPT by showing that it is possible to
+separate the gauge-independent dynamics of perturbation from the issues of gauge-fixing and space-
+time reconstruction. In particular we showed how the spacetime reconstruction can be pursued
+with the sole knowledge of gauge-fixing conditions. Moreover the discussed Kucaˇr decomposition
+serves as a useful and insightful tool to the study of gauge-fixing conditions and spacetime recon-
+struction. The space of gauge-fixing conditions and the formula for the spacetime reconstruction
+is given explicitly for any gauge.
+This approach might be applied to multiple conceptual problems in quantum cosmology, such
+as the time problem, the semi-classical spacetime reconstruction , or the relation between the
+kinematical and reduced phase space quantization. Moreover, the complete control over the gauge-
+fixing issue provided by the presented method, could be very useful for the problem of gluing
+perturbed spacetimes to other spacetime models (e.g., ones that include non-linearities).
+The
+choice of the gluing surface and its coordinates should be nicely described by our method.
+4 The spacetime coordinates system is independent from the evolution of this variables.
+
+7
+ACKNOWLEDGMENTS
+The author acknowledge the support of the National Science Centre (NCN, Poland) under the
+research grant 2018/30/E/ST2/00370.
+[1] P.
+Ma�lkiewicz,
+Class.
+Quant.
+Grav.
+36
+(2019)
+no.21,
+215003
+doi:10.1088/1361-6382/ab45aa
+[arXiv:1810.11621 [gr-qc]].
+[2] A. Boldrin and P. Ma�lkiewicz, Class. Quant. Grav. 39 (2022) no.2, 025005 doi:10.1088/1361-6382/ac3bda
+[arXiv:2105.05325 [gr-qc]].
+[3] K. A. Malik and D. R. Matravers, Gen. Rel. Grav. 45 (2013), 1989-2001 doi:10.1007/s10714-013-1573-2
+[arXiv:1206.1478 [astro-ph.CO]].
+[4] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78 (1984), 1-166 doi:10.1143/PTPS.78.1
+[5] P. A. M. Dirac, “Lectures on quantum mechanics,” ISBN:9780486417134.
+[6] K. Kuchaˇr , Journal of Mathematical Physics doi:10.1063/1.1666050.
+[7] A. Boldrin and P. Ma�lkiewicz, Class. Quant. Grav. 40 (2023) no.1, 015003 doi:10.1088/1361-6382/aca385
+[arXiv:2206.06926 [gr-qc]].
+[8] R. Arnowitt, S. Deser, C. W. Misner, “Dynamical Structure and Definition of Energy in General Rela-
+tivity,” Phys. Rev., Vol. 116, Issue 5, p. 1322-1330, 1959 doi:10.1103/PhysRev.116.1322
+

-tFAT4oBgHgl3EQfqR1A/content/tmp_files/load_file.txt

@@ -0,0 +1,181 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf,len=180
+page_content='The Gauge issue and the Hamiltonian theory of cosmological perturbations  Alice Boldrin1, ∗  1National Centre for Nuclear Research, Pasteura 7, 02-093 Warszawa, Poland  We present a general formalism for the Hamiltonian description of perturbation theory  around any spatially homogeneous spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' We employ and refine the Dirac method for  constrained systems, which is very well-suited to cosmological perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' This approach  includes a discussion of the gauge-invariant dynamics of perturbations as well as an analysis  of gauge transformations, gauge-fixing, partial gauge-fixing and spacetime reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  We will introduce the Kuchaˇr parametrization of the kinematical phase space as a conve-  nient tool for studying the gauge transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The key element of this approach is the  reconstruction of spacetime based on gauge-fixing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  INTRODUCTION  In the attempt to obtain a quantum theory suitable for the description of the primordial struc-  ture of the Universe, we study the Hamiltonian formalism for cosmological perturbation theory  (CPT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' This work has been done before with different background spacetime models like the Fried-  man universe [1] and the Bianchi Type I model [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Our aim is to study the complete Hamiltonian  formalism in a general background focusing on the gauge independent description of CPT as well  as the issue of gauge fixing (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' [3, 4] for alternative discussions on the gauge issue in CPT),  gauge transformations and spacetime reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' We employ the Dirac method [5] to study the  Hamiltonian in different gauges and reconstruct the spacetime metric from gauge-invariant quan-  tities (Dirac observables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' We also discuss an alternative method based on the so-called Kuchaˇr  decomposition [6] which provides a parametrization of the phase space in which the constrains  play the role of canonical variables conjugate to the gauge-fixing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' For a more detailed  discussion and for an application of the presented method, see [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  ∗ Alice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='Boldrin@ncbj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='gov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='pl  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='08646v1  [gr-qc]  20 Jan 2023  2  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  COSMOLOGICAL PERTURBATION THEORY  The Hamiltonian in the the Arnowitt-Deser-Misner (ADM) formalism [8] expanded to second  order reads1  H =  �  T3  �  NH(0)  0  + NH(2)  0  + δN µδHµ  �  d3x  (1)  where N is the background lapse function and δN µ, with µ = 0, i, are the first order lapse and  shift functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The Hamiltonian densities H(0) and H(2) are respectively zeroth and second order,  whereas δHµ represent the first order constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' We assume a spatially homogeneous background  spacetime with spatial coordinates defined such that the background shift vector Ni vanishes as  well as the background Hamiltonian H(0)  i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The Hamiltonian (1) is a function of the background  canonical variables ¯qij and ¯πij which are respectively the three-metric and three-momenta, and the  perturbed variables defined as δqij = qij − qij and δπij = πij − πij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  The Hamiltonian (1) defines a gauge for the following reasons:  First, at each spatial point the constraints algebra is closed, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  {δHi, δHj} = 0,  {δH0, δHi} = 0,  (2)  where this result is true for any homogeneous background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Furthermore the constraints are dy-  namically stables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  {H, δH0} = −δHi  ,i(x) ≈ 0,  {H, δHi} = 0,  (3)  where the ”weak equality” ≈, means that the equality holds in the constraint surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  GAUGE-FIXING AND DIRAC PROCEDURE  The four constraints δHµ generate a gauge freedom which can be removed by imposing four  gauge-fixing conditions δcµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The Poisson bracket between the gauge-fixing conditions and the  constraints form an invertible matrix det{δcµ, δHµ} ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Applying the constraints and the gauge-  fixing conditions we can reduce our Hamiltonian which will now depend on 4 physical variables  (δqphys  I  , δπI  phys) instead of the 12 ADM perturbation2 variables (δqij, δπij).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Those new variables  form a canonical coordinate system on the submanifold in the kinematical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  This  1 We assume the topology of the spacetime to be M ≃ T3 × R so to have a spatially compact universe and avoid  ambiguous definitions of the symplectic structure for background (homogeneous) variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  2 We assume the vacuum case for the sake of clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' See [1] or [2] for the Dirac method applied when there is matter  content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  3  submanifold is thus called the physical phase space3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  The parametrization provided by these  physical variables is defined by the gauge-fixing surface that intersects all gauge orbits (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' 1  ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' It is convenient to define a set of gauge-independent variables defined as  {δDI, δHµ} ≈ 0, ∀µ,  (4)  which parametrize the space of gauge orbits in the constraints surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Those variables are known  as Dirac observables and are equal to the number of physical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' There exists a one-to-one  correspondence between the Dirac observables and the physical variables, such that  δDI + ϵµ  I δcµ + ξµ  I δHµ = δOphys  I  (δqphys  I  , δπI  phys)  (5)  where ϵµ  I and ξµ  I are background coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Using this new parametrization the Hamiltonian can  be written in a gauge-independent manner as H(2)  phys = H(2)  red + H(2)  ext, where H(2)  phys denotes the so  called physical Hamiltonian, H(2)  red is the reduced Hamiltonian in terms of the physical variables and  H(2)  ext is the extra Hamiltonian generated by the time-dependent canonical transformation needed  to change parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Graphical representation of the Dirac procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  SPACETIME RECONSTRUCTION  In the previous section we discussed how to obtain the physical Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  In order to  reconstruct the spacetime we still need to find the values of the first-order lapse and shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' To do  so we use the consistency equation {δcµ, H} = 0, which, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' (1), implies  δN µ  N  = −{δcν, δHµ}−1 �  {δcν, δH(0)} + {δcν, H(2)}  �  (6)  3 It’s canonical structure is now given by the Dirac brackets {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' }D = {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='} − {.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=', δφµ}{δφµ, δφν}−1{δφν, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' }, where  δφµ ∈ (δHµ, δcµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  4  This equation is only meaningful in the constraint surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  KUCHAˇR DECOMPOSITION  We present a different parametrization of the kinematical phase space where the constraints  take the role of canonical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' For instance, we define two sets of canonical variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The  first set comprises the first order constraints δHµ and the 4 gauge-fixing functions, here denoted  as δCµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The second pair of canonical variables is given by the Dirac observables δDI, defined in  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The Hamiltonian written in this parametrization will then be  H → HK = H + K =  � �  NH(0)  0  + N(H(2)  0  + K) + δN µδHµ  �  d3x,  (7)  where K is the extra Hamiltonian coming from the time-dependent parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' We notice  that, since the constraints are conserved in the constraint surface, terms of the form ∝ δCµδCν,  ∝ δQIδCµ and ∝ δPIδCµ are not present in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Moreover, considering that H(2) ≈ H(2)  red  and K(2) ≈ H(2)  ext, which tells us the two dynamics must be weakly equal, we have that the total  Hamiltonian can only be of the form  HK =N  � �  H(2)  phys(δQI, δP I)  �  ��  �  physical part  +  +  �  λµI  1 δQI + λµ  2IδPI + λµν  3 δHν + λµ  4νδCν + δN µ  N  �  Hµ  �  ��  �  weakly vanishing part  �  d3x,  (8)  where λµI  1 , λµ  I2 and λµν  3  are zeroth-order coefficients that can depend on the gauge-fixing δCµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  The value of λµ  4ν is gauge-invariant, it is showed to be fixed unambiguously by the algebra of the  hypersurface (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  Gauge transformations  An interesting property of the Kuchaˇr decomposition comes from the freedom in the choice of  the canonical variable δCµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' This means that we have a class of parametrizations of the kinematical  phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' In particular, we can define the new set of gauge-fixing conditions as δ ˜Cµ, and the full  gauge transformation will be given by the map G : (δHµ, δCµ, δQI, δP I) → (δ ˜Hµ, δ ˜Cµ, δ ˜QI, δ ˜P I),  where δHµ = δ ˜Hµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' We are free to assume that the new gauge-fixing functions are thus canonically  conjugate to the constraints δHµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Thus we have {δHν, δ ˜Cµ − δCµ} = 0, which implies  δ ˜Cµ = δCµ + αµ  I δP I + βµIδQI + γµνδHν,  (9)  5  where αµ  I , βµI and γµν are background parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  The gauge-fixing condition is only relevant in the constraints surface, so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' (9) is fully deter-  mined by the parameters αµ  I and βµI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Moreover it means that the space of gauge-fixing conditions  is the affine space of dimension equal to the number od Dirac observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The introduction of a  different gauge will lead to the new Hamiltonian H ˜  K, with an extra Hamiltonian density ∆K(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  Studying the new symplectic form of the system we find that γµν depends only on αµ  I and βµI,  which thus are the only parameters needed to uniquely determine the gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  Spacetime reconstruction  As discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' IV, the spacetime reconstruction is obtained by the dynamical equations  δ ˙Cµ = 0, which means that it is sensitive to the chosen parametrization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' In particular in the  Kuchaˇr parametrization we will have {δCν, HK}K = 0, which, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' (7) becomes  δN µ  N  = −∂(H(2) + K(2))  ∂δHµ  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  (10)  Notice the above formula only depends on the weakly vanishing part of the Hamiltonian since the  lapse and shift are gauge-dependent quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' It is interesting to consider the difference between  the lapse and shift in two gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' (8), we have  δ ˜Nµ  N  ����  δ ˜Cµ=0  − δN µ  N  ����  δCµ=0  ≈  ≈  �  �λµ  4νβνI + ˙βµI +  ∂2H(2)  phys  ∂δQI∂δP J βµJ −  ∂2H(2)  phys  ∂δQI∂δQJ  αµ  J  �  � δQI  +  �  �λµ  4ναν  I + ˙αµ  I −  ∂2H(2)  phys  ∂δP I∂δQJ  αµ  J +  ∂2H(2)  phys  ∂δP I∂δP J βµJ  �  � δP I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  (11)  We see that the spacetime reconstruction in a new gauge can be obtained by the lapse and shift in  the initial gauge plus some terms which solely depend on the physical part of the Hamiltonian H(2)  phys  and the gauge-invariant coefficient λµ  4ν, which can be obtained from the algebra of the hypersurface  deformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  PARTIAL GAUGE-FIXING  We previously discussed the gauge-fixing defined as setting the conditions δCµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' However it  can be interesting to study the case in which these 4 conditions are substituted with conditions on  6  the lapse and shift functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' This is what we call partial gauge-fixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' From this consideration we  can study the transformations which preserve the lapse and shift functions, that is, δ ˜  Nµ  N  ��  δ ˜Cµ=0 −  δNµ  N  ��  δCµ=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' (11) and solving it for αν  I and βµI, we can solve the ambiguity in the  choice of the gauge-fixing condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  ˙αµ  I = −βµJ  ∂2H(2)  phys  ∂δP J∂δP I + αµ  J  ∂2H(2)  phys  ∂δQJ∂δP I − λµ  4ναν  I,  ˙βµI = −βµJ  ∂2H(2)  phys  ∂δP J∂δQI  + αµ  J  ∂2H(2)  phys  ∂δQJ∂δQI  − λµ  4νβνI,  (12)  The above equations fix the gauge-fixing function at all times once δCµ(t0) is fixed at an initial  time t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' This means that the choice of δCµ(t0) fixes the initial three-surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Given the initial  values of the Dirac observables (δQI(t0), δP I(t0)), we are able to explicitly reconstruct the initial  three-surface in terms of the ADM perturbation variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Moreover we are able to fully reconstruct  the spacetime geometry since the evolution of the three-surface with its coordinates is completely  determined by the evolution of the gauge-fixing function δ ˜Cµ(t) and the independent evolution of  the gauge-invariant variables4 (δQI(t), δP(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  CONCLUSIONS  We were able to simplify the Hamiltonian approach to CPT by showing that it is possible to  separate the gauge-independent dynamics of perturbation from the issues of gauge-fixing and space-  time reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' In particular we showed how the spacetime reconstruction can be pursued  with the sole knowledge of gauge-fixing conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Moreover the discussed Kucaˇr decomposition  serves as a useful and insightful tool to the study of gauge-fixing conditions and spacetime recon-  struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' The space of gauge-fixing conditions and the formula for the spacetime reconstruction  is given explicitly for any gauge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  This approach might be applied to multiple conceptual problems in quantum cosmology, such  as the time problem, the semi-classical spacetime reconstruction , or the relation between the  kinematical and reduced phase space quantization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Moreover, the complete control over the gauge-  fixing issue provided by the presented method, could be very useful for the problem of gluing  perturbed spacetimes to other spacetime models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=', ones that include non-linearities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  The  choice of the gluing surface and its coordinates should be nicely described by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  4 The spacetime coordinates system is independent from the evolution of this variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  7  ACKNOWLEDGMENTS  The author acknowledge the support of the National Science Centre (NCN, Poland) under the  research grant 2018/30/E/ST2/00370.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  Ma�lkiewicz,  Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  36  (2019)  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='21,  215003  doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1088/1361-6382/ab45aa  [arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='11621 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Boldrin and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Ma�lkiewicz, Class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' 39 (2022) no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='2, 025005 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1088/1361-6382/ac3bda  [arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='05325 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  [3] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Malik and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Matravers, Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Grav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' 45 (2013), 1989-2001 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1007/s10714-013-1573-2  [arXiv:1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1478 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='CO]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  [4] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Kodama and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Sasaki, Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' 78 (1984), 1-166 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1143/PTPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
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+page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Dirac, “Lectures on quantum mechanics,” ISBN:9780486417134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
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+page_content='1666050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
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+page_content=' Quant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
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+page_content='1, 015003 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1088/1361-6382/aca385  [arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='06926 [gr-qc]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='  [8] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Arnowitt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Deser, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Misner, “Dynamical Structure and Definition of Energy in General Rela-  tivity,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=', Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' 116, Issue 5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content=' 1322-1330, 1959 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1103/PhysRev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}
+page_content='1322' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tFAT4oBgHgl3EQfqR1A/content/2301.08646v1.pdf'}

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@@ -0,0 +1,2635 @@
+arXiv:2301.12137v1  [quant-ph]  28 Jan 2023
+Unitarily inequivalent local and global Fourier transforms in multipartite
+quantum systems
+C. Lei and A. Vourdas∗
+Department of Computer Science,
+University of Bradford,
+Bradford BD7 1DP, United Kingdom
+[email protected]
+[email protected]
+Abstract. A multipartite system comprised of n subsystems, each of which is described with
+‘local variables’ in Z(d) and with a d-dimensional Hilbert space H(d), is considered. Local Fourier
+transforms in each subsystem are defined and related phase space methods are discussed (displace-
+ment operators, Wigner and Weyl functions, etc). A holistic view of the same system might be
+more appropriate in the case of strong interactions, which uses ‘global variables’ in Z(dn) and a
+dn-dimensional Hilbert space H(dn). A global Fourier transform is then defined and related phase
+space methods are discussed. The local formalism is compared and contrasted with the global for-
+malism. Depending on the values of d, n the local Fourier transform is unitarily inequivalent or
+unitarily equivalent to the global Fourier transform. Time evolution of the system in terms of both
+local and global variables, is discussed. The formalism can be useful in the general area of Fast
+Fourier transforms.
+I.
+INTRODUCTION
+Entanglement and stronger than classical correlations in multipartite systems, are fundamental concepts in
+quantum mechanics (e.g., [1]). Even if the various components of the system are physically located far from
+each other, strong correlations and strong interactions between them, weaken the concept of separate identity
+for each component . This motivates a comparison between the formalism of a multipartite system, with a
+holistic formalism of the same system that uses global quantities.
+We consider a finite quantum system with variables in Z(d) where d is an odd integer, described by the
+d-dimensional Hilbert space H(d) (e.g.[2, 3]). We also consider a multipartite system that consists of n of
+these systems (which are possibly located far from each other). In this system the positions and momenta
+take values in [Z(d)]n = Z(d) × ... × Z(d). The system is described with the dn-dimensional Hilbert space
+H = H(d) ⊗ ... ⊗ H(d).
+In the case of strong correlations and strong interactions between the n components we introduce a holistic
+approach and regard this as one system with variables in Z(dn) and dn-dimensional Hilbert space H(dn). We
+note that
+• The Hilbert space H is isomorphic to the H(dn), because they both have the same dimension.
+• There is a bijective map between the sets [Z(d)]n and Z(dn) given below in Eq.(35) (in fact we can have
+many bijective maps between these two sets). However the [Z(d)]n as a ring is not isomorphic to the ring
+Z(dn) (see Eq.(36) below).
+With this in mind, we study the following:
+• We define a local Fourier transform FL in the phase space [Z(d)]n ×[Z(d)]n of the system when considered
+as n-component system. We also define a global Fourier transform FG in the phase space Z(dn) × Z(dn)
+of the system when considered as a single system. This has been introduced briefly in a different context
+in ref.[4], and here it is studied as a problem in its own right and in connection with a global phase
+space formalism. We show that depending on the values of d, n the local Fourier transform is unitarily
+inequivalent (unitarily equivalent) to the global Fourier transform. By that we mean that there exists no
+unitary transformation U (there exists such a transformation U) so that FG = UFLU †. This is discussed
+in section IV D and in proposition IV.4.
+
+2
+• Starting from an orthonormal basis of ‘position states’, we use local and global Fourier transforms to
+define local and global momentum states. Some of the local momentum states are the same as the global
+momentum states as discussed in proposition IV.3. We also define local position and momentum operators,
+and also global position and momentum operators. We do numerical calculations of the time evolution
+for the case where the Hamiltonian is expressed in terms of local variables and also for the case where the
+Hamiltonian is expressed in terms of global variables (section V B). For multipartite systems with strong
+interactions between the various components, it might be more appropriate to express the Hamiltonian in
+terms of the global variables.
+• We define a local phase space formalism in [Z(d)]n × [Z(d)]n and a global phase space formalism in
+Z(dn) × Z(dn). Displacements, Wigner and Weyl functions, etc, are defined in these two cases. Density
+matrices which have only diagonal elements with respect to the position basis, have the same local and
+global Wigner function (proposition V.3). The difference between local and global Wigner functions, is
+contained entirely in the off-diagonal elements.
+• Deviations of a density matrix ρ from the corresponding factorisable density matrix R(ρ) (defined in
+Eq.(16)) are described with the matrices RL, �RL and RG, �RG. They describe classical and quantum
+correlations in the multipartite system described by ρ (section V E).
+• Understanding of the relationship between global and local Fourier transforms and related phase space
+methods, might be useful in other areas like fast Fourier transforms. For n = 2 we show that the global
+Fourier transform can be expressed in terms of many local Fourier transforms (section IV E). This is
+similar to the Cooley-Tukey formalism in fast Fourier transforms[5–7]. The general area of Fast Fourier
+transforms (in a quantum or even classical context) is a potential application of the present formalism.
+• In the case that the local and global Fourier transform are unitarily inequivalent (Eq.(65) below), the
+concept of a multipartite system (and related concepts like entanglement) is fundamentally different from
+that of a single quantum system. But if they are unitarily equivalent (Eq.(64) below), the distinction
+between a multipartite system and a single system is weak. Unitary equivalence means that with a change
+of basis one concept is transformed to another, and consequently there is no fundamental difference
+between the two. In this case, further work is needed in order to clarify the correspondence between the
+two (especially of entanglement which is a concept applicable to a multipartite system but not to a single
+system).
+In section 2 we review briefly the phase-space formalism for systems with finite Hilbert space[2, 3]. In section
+3 we apply this to each component of a n-partite system, and this is the ‘local formalism’. In section 4 we define
+the global Fourier transform and discuss for which values of d, n it is unitarily inequivalent to the local Fourier
+transform. In section 5 we present the global phase space formalism and compare and contrast it with the local
+formalism. In section 6, we present examples. We conclude in section 7 with a discussion of our results.
+II.
+BACKGROUND
+We consider a quantum system (qudit) with variables in the ring Z(d) of integers modulo d where d is an
+odd integer. H(d) is the d-dimensional Hilbert space describing this system. There are well known technical
+differences between quantum systems with odd dimension d and even dimension d (e.g., [8–10]). In this paper
+we consider systems with odd dimension d.
+Let |X; j⟩ where j ∈ Z(d) be an orthonormal basis in H(d). The X in the notation is not a variable, it simply
+indicates ‘position states’. The finite Fourier transform F is given by[11]
+F =
+1
+√
+d
+�
+j,k
+ωd(jk)|X; j⟩⟨X; k|;
+ωd(α) = exp
+�
+i2πα
+d
+�
+;
+α, j, k ∈ Z(d)
+F 4 = 1;
+FF † = 1.
+(1)
+
+3
+Its trace is[2]
+d = 4m + 1 → TrF = 1;
+d = 4m + 3 → TrF = i.
+(2)
+We act with F on position states and get the dual basis
+|P; j⟩ = F|X; j⟩.
+(3)
+The P in the notation is not a variable, it simply indicates ‘momentum states’.
+Using the relation
+1
+d
+�
+k
+ωd[(j + ℓ)k] = δ(j, −ℓ),
+(4)
+we show that F 2 is the parity operator around the origin:
+F 2 = 1
+d
+�
+j,k,ℓ
+ωd[(j + ℓ)k]|X; j⟩⟨X; ℓ| =
+�
+j
+|X; j⟩⟨X; −j|.
+(5)
+The phase space of this system is Z(d) × Z(d) and in it we introduce the displacement operators
+Xβ =
+�
+j
+ωd(−jβ)|P; j⟩⟨P; j| =
+�
+j
+|X; j + β⟩⟨X; j|;
+Zα =
+�
+j
+|P; j + α⟩⟨P; j| =
+�
+j
+ωd(αj)|X; j⟩⟨X; j| = FXαF †;
+Xd = Zd = 1;
+XβZα = ZαXβωd(−αβ);
+α, β ∈ Z(d).
+(6)
+General displacement operators are the unitary operators
+D(α, β) = ZαXβωd(−2−1αβ);
+[D(α, β)]† = D(−α, −β);
+D(α1, β1)D(α2, β2) = D(α1 + α2, β1 + β2)ωd[2−1(α1β2 − α2β1)].
+(7)
+The 2−1 = d+1
+2
+is an integer in Z(d) with odd d, considered here. The D(α, β)ω(γ) form a representation of the
+Heisenberg-Weyl group. We note that
+D(α, β)|X; j⟩ = ωd(2−1αβ + αj)|X; j + β⟩;
+D(α, β)|P; j⟩ = ωd(−2−1αβ − βj)|P; j + α⟩.
+(8)
+The
+X = −i d
+2π log(Z);
+P = i d
+2π log(X);
+FPF † = −X.
+(9)
+are d × d matrices which can be interpreted as position and momentum operators. The commutator [X, P] can
+be calculated (it is not i1) but it has no mathematical significance because the Heisenberg-Weyl group in this
+context is discrete, and the concept of generators is non-applicable. Hamiltonians can be written as functions
+of these operators as h(X, P).
+
+4
+A.
+Wigner and Weyl functions
+The parity operator (around the point (γ, δ)) is defined as
+P({γ, δ}) = D(γ, δ)F 2[D(γ, δ)]†;
+[P({γ, δ})]2 = 1.
+(10)
+It is related to the displacement operators through the Fourier transform
+P(γ, δ) = 1
+d
+�
+α,β
+D(α, β)ωd(βγ − αδ);
+D(α, β) = 1
+d
+�
+γ,δ
+P(γ, δ)ωd(−βγ + αδ).
+(11)
+If ρ is a density matrix, we define the Wigner function W(γ, δ) and the Weyl function �
+W(α, β) as:
+W(γ, δ) = Tr[ρP(γ, δ)];
+�
+W(α, β) = Tr[ρD(α, β)].
+(12)
+From Eq.(11) follows immediately that they are related to each other through the Fourier transform:
+W(γ, δ) = 1
+d
+�
+α,β
+�
+W(α, β)ωd(βγ − αδ);
+�
+W(α, β) = 1
+d
+�
+γ,δ
+W(γ, δ)ωd(−βγ + αδ).
+(13)
+The following marginal properties of the Wigner function are well known for odd values of the dimension d
+(e.g., [2]):
+1
+d
+�
+γ
+W(γ, δ) = ⟨X; δ|ρ|X; δ⟩;
+1
+d
+�
+δ
+W(γ, δ) = ⟨P; γ|ρ|P; γ⟩;
+1
+d
+�
+γ,δ
+W(γ, δ) = 1.
+(14)
+III.
+LOCAL PHASE SPACE METHODS
+A.
+Local Fourier transforms
+We consider a n-partite system comprised of n components each of which is a qudit. This system is described
+with the dn-dimensional Hilbert space H = H(d) ⊗ ... ⊗ H(d). Positions and momenta take values in [Z(d)]n =
+Z(d) × ... × Z(d). If ρ is the density matrix of the system, we use the notation
+˘ρr = Tri̸=rρ;
+r = 0, ..., n − 1,
+(15)
+for the reduced density matrix describing the r-component of the system. We also define the corresponding
+factorisable density matrix
+R(ρ) = ˘ρ0 ⊗ ... ⊗ ˘ρn−1;
+TrR(ρ) = 1,
+(16)
+
+5
+and the correlator
+C(ρ) = ρ − R(ρ);
+TrC(ρ) = 0.
+(17)
+For factorisable density matrices R(ρ) = ρ and C(ρ) = 0. Below we compare quantities for ρ with the corre-
+sponding quantities for R(ρ).
+We consider the basis
+|X; j0, ..., jn−1⟩ = |X; j0⟩ ⊗ ... ⊗ |X; jn−1⟩;
+jr ∈ Z(d).
+(18)
+called basis of position states. We also consider the local Fourier transforms:
+FL = F ⊗ ... ⊗ F;
+F 4
+L = 1;
+FLF †
+L = 1.
+(19)
+The index L in the notation stands for local. Acting with FL on the basis |X; j0, ..., jn−1⟩ we get the ‘local
+momentum states’:
+|PL; j0, ..., jn−1⟩ = FL|X; j0, ..., jn−1⟩ =
+1
+√
+dd
+n−1
+�
+r=0
+� d−1
+�
+kr=0
+ωd(jrkr)|X; kr⟩
+�
+= |P; j0⟩ ⊗ ... ⊗ |P; jn−1⟩.
+(20)
+F 2
+L is a parity operator in the sense that
+F 2
+L|X; j0, ..., jn−1⟩ = |X; −j0, ..., −jn−1⟩.
+(21)
+For later use we define the matrix elements of the correlator C(ρ):
+C(X; j0, ..., jn−1) = ⟨X; j0, ..., jn−1|C(ρ)|X; j0, ..., jn−1⟩
+= ⟨X; j0, ..., jn−1|ρ|X; j0, ..., jn−1⟩ −
+n−1
+�
+r=0
+⟨X; jr|˘ρr|X; jr⟩,
+(22)
+and
+C(PL; j0, ..., jn−1) = ⟨PL; j0, ..., jn−1|C(ρ)|PL; j0, ..., jn−1⟩
+= ⟨PL; j0, ..., jn−1|ρ|PL; j0, ..., jn−1⟩ −
+n−1
+�
+r=0
+⟨P; jr|˘ρr|P; jr⟩.
+(23)
+Then
+TrC(ρ) =
+�
+j0,...,jn−1
+C(X; j0, ..., jn−1) =
+�
+j0,...,jn−1
+C(X; j0, ..., jn−1) = 0.
+(24)
+For factorisable density matrices C(X; j0, ..., jn−1) = C(PL; j0, ..., jn−1) = 0.
+B.
+Displacements in [Z(d) × Z(d)]n
+The phase space of the system is [Z(d) × Z(d)]n and local displacement operators in it are defined as
+XL({βr}) =
+�
+jr
+ωd(−β0j0 − ... − βn−1jn−1)|PL; j0, ..., jn−1⟩⟨PL; j0, ..., jn−1|
+=
+�
+jr
+|X; j0 + β0, ..., jn−1 + βn−1⟩⟨X; j0, ..., jn−1|
+= Xβ0 ⊗ ... ⊗ Xβn−1,
+(25)
+
+6
+where r = 0, ..., n − 1, and
+ZL({αr}) =
+�
+jr
+|PL; j0 + α0, ..., jn−1 + αn−1⟩⟨PL; j0, ..., jn−1|
+=
+�
+jr
+ωd(α0j0 + ... + αn−1jn−1)|X; j0, ..., jn−1⟩⟨X; j0, ..., jn−1|
+= Zα0 ⊗ ... ⊗ Zαn−1 = FLXL({αr})F †
+L.
+(26)
+Since ZL({αr})ZL({γr}) = ZL({αr +γr}), the ZL({αr}) form a representation of [Z(d)]n as an additive group
+The same is true for the XL({βr}). Also
+XL({βr})ZL({αr}) = ZL({αr})XL({βr})ωd[−(α0β0 + ... + αn−1βn−1)]
+[XL({βr})]d = [ZL({αr})]d = 1.
+(27)
+Using the notation
+{αr, βr} = {a0, ..., an−1, β0, ..., βn−1},
+(28)
+general local displacement operators are defined as
+DL({αr, βr}) = ZL({αr})XL({βr})ωd[−2−1(α0β0 + ... + αn−1βn−1)]
+= D(α0, β0) ⊗ ... ⊗ D(αn−1, βn−1)
+αr, βr ∈ Z(d).
+(29)
+The DL({αr, βr})ω({γr}) form a representation of the Heisenberg-Weyl group of displacements in the phase
+space [Z(d) × Z(d)]n.
+The local parity operator (around the point {γr, δr} in the phase space [Z(d) × Z(d)]n) is defined as
+PL({γr, δr}) = DL({γr, δr})F 2
+L[DL({γr, δr})]† = P(γ0, δ0) ⊗ ... ⊗ P(γn−1, δn−1)
+[PL({γr, δr})]2 = 1.
+(30)
+It is related to the local displacement operators through the Fourier transform
+PL({γr, δr}) = 1
+dn
+�
+{αr,βr}
+DL({αr, βr})ωd
+�n−1
+�
+r=0
+(βrγr − αrδr)
+�
+;
+DL({αr, βr}) = 1
+dn
+�
+{γr,δr}
+PL({γr, δr})ωd
+�n−1
+�
+r=0
+(−βrγr + αrδr)
+�
+.
+(31)
+The proof of this follows easily from Eq.(11).
+C.
+Local Wigner and local Weyl functions in [Z(d) × Z(d)]n
+If ρ is a density matrix, we define the local Wigner function WL({γr, δr}|ρ) and the local Weyl function
+�
+WL({αr, βr}|ρ) as:
+WL({γr, δr}|ρ) = Tr[ρPL({γr, δr})];
+�
+WL({αr, βr}|ρ) = Tr[ρDL({αr, βr})].
+(32)
+From Eq.(31) follows immediately that they are related to each other through the Fourier transform:
+WL({γr, δr}|ρ) = 1
+dn
+�
+{αr,βr}
+�
+WL({αr, βr}|ρ)ωd
+�n−1
+�
+r=0
+(βrγr − αrδr)
+�
+;
+�
+WL({αr, βr}|ρ) = 1
+dn
+�
+{γr,δr}
+WL({γr, δr}|ρ)ωd
+�n−1
+�
+r=0
+(−βrγr + αrδr)
+�
+.
+(33)
+
+7
+IV.
+GLOBAL FOURIER TARNSFORMS
+A.
+A bijective map between the non-isomorphic rings [Z(d)]n and Z(dn)
+We consider a bijective map between [Z(d)]n and Z(dn) as follows. We first take each jr ∈ Z(d) and �j ∈ Z(dn)
+in the ‘periods’
+�
+−d − 1
+2
+, d − 1
+2
+�
+;
+�
+−dn − 1
+2
+, dn − 1
+2
+�
+,
+(34)
+correspondingly (for odd d). We introduce the bijective map
+j = (j0, ..., jd−1) ↔ �j = j0 + j1d + ... + jn−1dn−1.
+(35)
+We then take each jr modulo d and the �j modulo dn, and we get a bijective map from [Z(d)]n to Z(dn). Numbers
+in Z(dn) will be denoted with a ‘hat’, so that it is clear whether a number belongs to Z(d) or to Z(dn).
+The Hilbert space H is isomorphic to H(dn) (a dn-dimensional Hilbert space describing systems with variables
+in Z(dn)). But the [Z(d)]n as a ring (with addition and multiplication componentwise), is not isomorphic to the
+ring Z(dn) because addition and multiplication is different, and consequently our ‘local formalism’ is different
+from our ‘global formalism’. Indeed
+�j + �k ̸= �
+j + k;
+�j�k ̸= �
+jk.
+(36)
+The sum is different because �j +�k in Z(dn) has the ‘carry’ rule and the r-component might be jr +kr +1 rather
+than jr + kr . In contrast, there is no ‘carry’ rule in [Z(d)]n:
+j + k = (j0 + k0, ..., jd−1 + kd−1) ↔ �
+j + k = (j0 + k0) + (j1 + k1)d + ... + (jn−1 + kn−1)dn−1.
+(37)
+Also multiplication in Z(dn) is
+�j�k = j0k0 + d(j1k0 + k1j0) + ... + dn−1(j0kn−1 + ... + jn−1k0).
+(38)
+The corresponding multiplication in [Z(d)]n is
+(j0, ..., jn−1) · (k0, ..., kn−1) = (j0k0, ..., jn−1kn−1),
+(39)
+and with the bijective map in Eq.(35) this corresponds to
+�
+jk = j0k0 + d(j1k1) + ... + dn−1(jn−1kn−1).
+(40)
+It is seen that in general �j�k ̸= �
+jk (but �1�k = �k).
+Example IV.1. We consider the elements of Z(3) in the ‘period’ [−1, 1] and the elements of Z(9) in the ‘period’
+[−4, 4]. A bijective map between [Z(3)]2 and Z(9) is as follows
+�
+(−1, −1) = �
+−4;
+�
+(0, −1) = �
+−3;
+�
+(1, −1) = �
+−2;
+�
+(−1, 0) = �
+−1;
+�
+(0, 0) = �0;
+�
+(1, 0) = �1;
+�
+(−1, 1) = �2;
+�
+(0, 1) = �3;
+�
+(1, 1) = �4.
+(41)
+An example of addition that confirms Eq.(36) is the following:
+�
+(1, 1) + �
+(1, 1) = �4 + �4 = �
+−1;
+�
+(1, 1) + (1, 1) =
+�
+(−1, −1) = �
+−4.
+(42)
+An example of multiplication that confirms Eq.(36) is the following:
+�
+(−1, 0) · �
+(0, 1) = �
+−1 · �3 = �
+−3;
+�
+(−1, 0) · (0, 1) = �
+(0, 0) = �0.
+(43)
+
+8
+Remark IV.2. If d1, ..., dn are coprime to each other, then the Z(d1)×...×Z(dn) is isomorphic to Z(d1 ×...×dn).
+We can define a bijective map
+(j0, ..., jn−1) ↔ j;
+jr ∈ Z(dr);
+j ∈ Z(d1 × ... × dn),
+(44)
+such that
+(j0 + k0, ..., jn−1 + kn−1) ↔ j + k;
+(j0k0, ..., jn−1kn−1) ↔ jk.
+(45)
+This is based on the Chinese remainder theorem, and has been used by Good [12] in the context of fast Fourier
+transforms (see also [5–7]). In a quantum context it has been use in [2, 13] for factorisation of a quantum
+system into subsystems. Here we consider the case d1 = ... = dn and then the bijective map of Eq.(35) does
+not establish an isomorphism between the ring [Z(d)]n and the ring Z(dn) (because of Eq.(36)).
+B.
+Dual notation
+We use the following dual notation for position states, based on the bijective map in Eq.(35):
+|X; j0, ..., jn−1⟩ = |X;�j⟩.
+(46)
+When local operators act on them we use addition and multiplication in [Z(d)]n, in connection with the phase
+space [Z(d) × Z(d)]n. When global operators (defined below) act on them we use addition and multiplication
+in Z(dn), in connection with the phase space Z(dn) × Z(dn).
+Analogous dual notation is used for all quantities. For example, the displacement operators in Eq.(29) can
+be denoted as
+DL({αr, βr}) = DL(�α, �β).
+(47)
+In some equations both notations appear together.
+C.
+Global Fourier transforms
+The global Fourier transform in H is defined as:
+FG =
+1
+√
+dn
+�
+�j,�k
+ωdn(�j�k)|X; j0, ..., jn−1⟩⟨X; k0, ..., kn−1|.
+(48)
+The index G in the notation stands for global. It is easily seen that
+F 4
+G = 1;
+FGF †
+G = 1;
+FG ̸= FL.
+(49)
+Acting with FG on the basis |X; j0, ..., jn−1⟩ (which we also denote as |X;�j⟩) we get the ‘global momentum
+states’:
+|PG;�j⟩ = |PG; j0, ..., jn−1⟩ = FG|X;�j⟩ =
+1
+√
+dn
+n−1
+�
+r=0
+� d−1
+�
+kr=0
+ωdn[(j0dr + .. + jn−r−1dn−1)kr]|X; kr⟩
+�
+. (50)
+In the states |PG; j0, ..., jn−1⟩, the coefficients ωdn[(j0dr + .. + jn−r−1dn−1)kr] in the r-component depend on all
+j0, ..., jn−1, and the term ‘global’ refers to this. Information from all components is needed, in order to determine
+
+9
+these coefficients. In the local Fourier transform of Eq.(21), the coefficients ωd(jrkr) in the r-component depend
+only on jr. We note that
+⟨PL; ℓ0, ..., ℓn−1|PG; j0, ..., jn−1⟩ = ⟨X; ℓ0, ..., ℓn−1|F †
+LFG|X; j0, ..., jn−1⟩
+=
+1
+dn
+�
+�k
+ωdn(�j�k)ωd[−(ℓ0k0 + ... + ℓd−1kn−1)].
+(51)
+and that
+|⟨X; ℓ0, ..., ℓn−1|PL; j0, ..., jn−1⟩|2 = |⟨X; ℓ0, ..., ℓn−1|PG; j0, ..., jn−1⟩|2 = 1
+dn .
+(52)
+Proposition IV.3. We take the elements of Z(d) and the elements of Z(dn) in the ‘periods’ of Eq.(34). Then
+(1) The parity operator around the origin is the same in both the local and global formalism:
+F 2
+G = F 2
+L =
+
+
+
+
+
+0 · · · 0 1
+0 · · · 1 0
+...
+...
+...
+...
+1 · · · 0 0
+
+
+
+
+ .
+(53)
+Here the matrix is in the position basis.
+(2) For any n
+|PG; �
+−dn−1⟩ = |PL; −1, 0, ..., 0⟩;
+|PG; 0⟩ = |PL; 0, 0, ..., 0⟩;
+|PG; �
+dn−1⟩ = |PL; 1, 0, ..., 0⟩.
+(54)
+(3) For n = 2 we have the stronger result
+|PG; �
+dλ⟩ = |PL; λ, 0⟩;
+λ = −d − 1
+2
+, ..., d − 1
+2
+.
+(55)
+At least d of the global momentum states are equal to d of the local momentum states.
+Proof.
+(1) For Zdn Eq.(4) becomes
+1
+dn
+�
+�k
+ωdn[(�j + �ℓ)�k] = δ(�j + �ℓ, 0);
+�j, �ℓ ∈ Zdn.
+(56)
+The �j + �ℓ = 0 implies jr + ℓr = 0, and we prove that
+F 2
+G = 1
+dd
+�
+�j,�k,�ℓ
+ωdn[(�j + �ℓ)�k]|j0, ..., jn−1⟩⟨ℓ0, ..., ℓn−1| =
+�
+j0,...,jn−1
+|j0, ..., jn−1⟩⟨−j0, ..., −jn−1| = F 2
+L.
+(57)
+(2) Using Eq.(51) we get
+⟨PL; 1, ..., 0|PG; �
+dn−1⟩ =
+1
+dn
+�
+�k
+ωdn(�
+dn−1�k)ωd(−k0)
+=
+1
+dn
+�
+�k
+ωdn(dn−1k0)ωd(−k0) = 1
+dn
+�
+�k
+1 = 1.
+(58)
+In a similar way we prove that
+|PG; �
+−dn−1⟩ = |PL; −1, 0, ..., 0⟩.
+(59)
+
+10
+(3) Using Eq.(51) with n = 2 and �k = k0 + dk1 we get
+⟨PL; λ, 0|PG; �
+dλ⟩ =
+1
+d2
+�
+�k
+ωd2(�
+dλ�k)ωd(−λk0) = 1
+d2
+�
+�k
+ωd2(�
+dλ�k − dλk0)
+=
+1
+d2
+�
+�k
+ωd2(d2λk1) = 1.
+(60)
+dλ takes values between − d2−1
+2
+and d2−1
+2
+and consequently λ takes the values in Eq.(55).
+For later use we define the matrix elements of the correlator C(ρ):
+C(X;�j) = ⟨X;�j|C(ρ)|X;�j⟩;
+E(PG;�j) = ⟨PG;�j|C(ρ)|PG;�j⟩.
+(61)
+In both the ‘local formalism’ and the ‘global formalism’ the position states are the same and the momentum
+states are different. Consequently the E(PG;�j) is different from the corresponding C(PL; j0, ..., jn−1).
+Then
+TrC(ρ) =
+�
+�j
+C(X;�j) =
+�
+�j
+E(PG;�j) = 0.
+(62)
+For factorisable density matrices C(X;�j) = E(PG;�j) = 0.
+D.
+Unitarily inequivalent local and global Fourier transforms
+In this paper we use the following definition of unitary equivalence. Two square matrices A, B are called uni-
+tarily equivalent if there exists a unitary matrix U such that A = UBU †. Unitary equivalence is an equivalence
+relation, i.e., matrices which are unitarily inequivalent belong to different equivalence classes. It is known[14, 15]
+that two normal d × d matrices A, B are unitarily equivalent if and only if
+||A|| = ||B||;
+Tr(Aη) = Tr(Bη);
+η = 1, ..., d;
+||A|| =
+��
+i,j
+|Aij|2.
+(63)
+We note that Specht’s general theorem for unitary equivalence (e.g., [15]) reduces easily to the above criteria
+for the Fourier matrices which are unitary.
+Some authors call the above unitary similarity, and they use the term unitary equivalence for the case where
+there exist two unitary matrices U, V such that A = UBV †.
+Proposition IV.4. In an n-partite system that has Hilbert space with dimension dn, the dn × dn matrices FG,
+FL are unitarily equivalent in the cases
+d = 4m + 1;
+d = 4m + 3 and n = 4N;
+d = 4m + 3 and n = 4N + 1.
+(64)
+The matrices FG, FL are unitarily inequivalent in the cases
+d = 4m + 3 and n = 4N + 2;
+d = 4m + 3 and n = 4N + 3.
+(65)
+
+11
+Proof. The matrices FG and FL are unitary and therefore normal, and we use the criterion in Eq.(63). We first
+note that
+||FG|| = ||FL|| = dn.
+(66)
+We next compare Tr(F η
+G) with Tr(F η
+L) for η = 1, ..., dn. But
+η = 4ǫ → F η
+G = F η
+L = 1;
+η = 4ǫ + 1 → F η
+G = FG;
+F η
+L = FL;
+η = 4ǫ + 2 → F η
+G = F η
+L;
+η = 4ǫ + 3 → F η
+G = F †
+G;
+F η
+L = F †
+L.
+(67)
+Therefore if Tr(FG) = Tr(FL) the FG, FL are unitarily equivalent, and if Tr(FG) ̸= Tr(FL) the FG, FL are
+unitarily inequivalent.
+For an n-partite system with dimension dn we get TrFL = (TrF)n and using Eq.(2) we get
+d = 4m + 1 → TrFL = 1;
+d = 4m + 3 → TrFL = in.
+(68)
+For the TrFG if d = 4m + 1, the dn = (4m + 1)n = 4M1 + 1 and we get
+d = 4m + 1 → TrFG = 1.
+(69)
+If d = 4m + 3 we consider two cases where n is an even number and an odd number. For even n we get
+dn = (4m + 3)n = 4M2 + 1 and for odd n we find dn = (4m + 3)n = 4M3 + 3. Therefore
+d = 4m + 3 and n = even → TrFG = 1;
+d = 4m + 3 and n = odd → TrFG = i.
+(70)
+Comparison of Eq.(68) with Eqs(69),(70) proves the proposition.
+In the case of Eq.(64) there exists a dn × dn unitary matrix U such that FG = UFLU †, i.e.,
+ωdn(�i�j) =
+�
+�j,�ℓ
+U(�i, �k)U(�j, �ℓ)ωd(ℓ0k0 + ... + ℓd−1kn−1).
+(71)
+So if instead of the basis |X;�j⟩ we choose the basis U|X;�j⟩ as position states, then the local Fourier transform
+with respect to the new basis is the global Fourier transform with respect to the old basis FG = UFLU †. So in
+this case the global Fourier transform is not a new concept. However U is in general a global transformation
+(it cannot be written as U1 ⊗ ... ⊗ Un) and for this reason there is some merit in the study of the global Fourier
+transform even in this case.
+The case of Eq.(65) where global and local Fourier transforms are unitarily inequivalent is clearly the most
+interesting one. Then the global Fourier transform is a new concept. In any case, the formalism below is the
+same for both cases in Eqs(64), (65).
+E.
+The global Fourier transform in terms of local Fourier transforms and applications in Fast Fourier
+transforms
+The general idea of Fast Fourier transforms is to express the ‘large’ Fourier transform in a large Hilbert space,
+as an ‘appropriate’ combination of ‘small’ Fourier transforms in smaller Hilbert spaces. Performing the ‘small’
+Fourier transforms instead of the ‘large’ Fourier transform, is computationally beneficial. The general formalism
+of this paper can be helpful in this direction.
+
+12
+As an example, we express in this section the global Fourier transform in terms of many local Fourier
+transforms. This is similar to the Cooley-Tukey formalism in fast Fourier transforms[5–7]. We only consider
+the special case n = 2, and we do not discuss complexity issues. But we point out that understanding of the
+relationship between global and local Fourier transforms and related phase space methods, can be useful in
+other areas like fast Fourier transforms.
+For the case n = 2 we get
+ωd2(�j�k) = ωd2(j0k0)ωd(j0k1 + j1k0).
+(72)
+Let |s⟩ be a quantum state in H = H(d) ⊗ H(d) and s(k0, k1) = ⟨X; k0, k1|s⟩. Then
+⟨X; j0, j1|FG|s⟩ = 1
+d
+�
+k0,k1
+ωd2(�j�k)s(k0, k1)
+= 1
+d
+�
+k0
+ωd(j1k0)ωd2(j0k0)
+�
+k1
+ωd(j0k1)s(k0, k1)
+=
+1
+√
+d
+�
+k0
+ωd(j1k0) [ωd2(j0k0)�s(k0, j0)] ,
+(73)
+where
+�s(k0, j0) =
+1
+√
+d
+�
+k1
+ωd(j0k1)s(k0, k1).
+(74)
+In this way the Fourier transform in a d2-dimensional space reduces to two Fourier transforms in d-dimensional
+spaces.
+V.
+GLOBAL PHASE SPACE METHODS
+A.
+Global displacements in Z(dn) × Z(dn)
+The phase space is defined by the Fourier transform and for global Fourier transforms is Z(dn) × Z(dn).
+Displacement operators in it are defined as
+XG(�β) =
+�
+�j
+ωdn(−�j �β)|PG;�j⟩⟨PG;�j| =
+�
+�j
+|X;�j + �β⟩⟨X;�j|,
+(75)
+and
+ZG(�α) =
+�
+�j
+|PG;�j + �α⟩⟨PG;�j| =
+�
+�j
+ωdn(�α�j)|X;�j⟩⟨X;�j| = FGXG(�α)F †
+G.
+(76)
+Addition in Z(dn) is used in these two equations, in contrast to Eqs(26), (25) where we have addition in [Z(d)]n.
+The XL({βr}) in Eq.(25) can also be written as
+XL(�β) =
+�
+�j
+|X; �
+j + β⟩⟨X;�j|.
+(77)
+We have explained that �
+j + β ̸= �j + �β, and consequently Eqs(75), (77) are an example of the difference between
+the local and global formalism. Also the ZL({αr}) in Eq.(26) can also be written as
+ZL(�α) =
+�
+jr
+ωd(α0j0 + ... + αn−1jn−1)|X;�j⟩⟨X;�j|.
+(78)
+
+13
+Comparison of Eqs(76), (78) again shows the difference between the local and global formalism.
+Since ZG(�α)ZG(�γ) = ZG(�α + �γ), the ZG(�α) form a representation of Z(dn) as an additive group (which is not
+isomorphic to [Z(d)]n). The same is true for the XG(�β). Also
+XG(�β)ZG(�α) = ZG(�α)XG(�β)ωdn(−�α�β);
+[XG(�β)]dn = [ZG(�α)]dn = 1.
+(79)
+These relations should be compared and contrasted with Eq.(27) for the local formalism. Global displacement
+operators are defined as
+DG(�α, �β) = ZG(�α)XG(�β)ωdn(−2−1�α�β).
+(80)
+Here 2−1 = dn+1
+2
+is an element of Z(dn). The DG(�α, �β)ωdn(�γ) form a representation of the Heisenberg-Weyl
+group of displacements in the phase space Z(dn) × Z(dn). We note that
+DG(�α, �β)|X;�j⟩ = ωdn(2−1�α�β + �α�j)|X;�j + �β⟩;
+DG(�α, �β)|PG;�j⟩ = ωdn(−2−1�α�β − �β�j)|PG;�j + �α⟩.
+(81)
+Also
+DG(�α, �β)|PL;�j⟩ =
+1
+√
+dn
+�
+�k
+ωd(j0k0 + ... + jn−1kn−1)DG(�α, �β)|X; �k⟩
+=
+1
+√
+dn
+�
+�k
+ωd(j0k0 + ... + jn−1kn−1)ωdn(2−1�α�β + �α�k)|X; �k + �β⟩.
+(82)
+These relations should be compared and contrasted to
+DL(�α, �β)|X;�j⟩ = ωd[2−1(α0β0 + ... + αn−1βn−1) + (α0j0 + ... + αn−1jn−1)]|X; �
+j + β⟩;
+DL(�α, �β)|PL;�j⟩ = ωd[−2−1(α0β0 + ... + αn−1βn−1) − (β0j0 + ... + βn−1jn−1)]|PL; �
+j + α⟩.
+(83)
+Also
+DL(�α, �β)|PG;�j⟩ =
+1
+√
+dn
+�
+�k
+ωdn(�j�k)DL(�α, �β)|X; �k⟩
+=
+1
+√
+dn
+�
+�k
+ωdn(�j�k)ωd[2−1(α0β0 + ... + αn−1βn−1) + (α0k0 + ... + αn−1kn−1)]|X; �
+k + β⟩.
+(84)
+The global parity operator (around the point (�γ, �δ) in the phase space Z(dn) × Z(dn)) is
+PG(�γ, �δ) = DG(�γ, �δ)F 2
+G[DG(�γ, �δ)]†;
+�
+PG(�γ, �δ)
+�2
+= 1.
+(85)
+In analogy to Eq.(11) we find that the global parity operator is related to the global displacement operators
+through the Fourier transform
+PG(�γ, �δ) = 1
+dn
+�
+�α,�β
+DG(�α, �β)ωdn(�β�γ − �α�δ);
+DG(�α, �β) = 1
+dn
+�
+�γ,�δ
+PG(�γ, �δ)ωdn(−�β�γ + �α�δ).
+(86)
+
+14
+Example V.1. We consider the case d = 3 and n = 2. In this case the global Fourier transform is unitarily
+inequivalent to the local Fourier transform. We work in the ‘periods ’ of Eq.(34).
+Let |X; k0, k1⟩ be the basis of position states. The globally Fourier transformed basis is
+|PG; j0, j1⟩ = 1
+3
+�
+k0,k1
+ω9[j0k0 + 3(j1k0 + j0k1)]|X; k0, k1⟩.
+(87)
+The jr, kr take the values −1, 0, 1. The locally Fourier transformed basis is
+|PL; j0, j1⟩ = 1
+3
+�
+k0,k1
+ω3(j0k0 + j1k1)|X; k0, k1⟩.
+(88)
+Then
+⟨PL; ℓ0, ℓ1|PG; j0, j1⟩ = 1
+9{1 + ω9(3j0 + 9j1)ω3(−ℓ1) + ω9(j0 + 3j1)ω3(−ℓ0)
++ ω9(4j0 + 12j1)ω3(−ℓ0 − ℓ1)]}.
+(89)
+We next consider the local displacement operator XL(−1, 1) which acts on the states |X; 1, 0⟩ and |PL; 1, 0⟩
+as follows:
+XL(−1, 1)|X; 1, 0⟩ = |X; 0, 1⟩,
+(90)
+and
+XL(−1, 1)|PL; 1, 0⟩ = ω3(1)|PL; 1, 0⟩.
+(91)
+XL(−1, 1) acts on the state |PG; 1, 0⟩ as follows:
+XL(−1, 1)|PG; 1, 0⟩ = XL(−1, 1)
+�
+j1,j0
+|X; j0, j1⟩⟨X; j0, j1|PG; 1, 0⟩
+= 1
+3
+�
+j1,j0
+ω9( �
+j0 + 3j1)|X; j0 − 1, j1 + 1⟩.
+(92)
+We also consider the corresponding global displacement operator XG(
+�
+−1 + 3 · 1) = XG(�2) which acts on the
+states |X; 1, 0⟩ = |X;�1⟩ and |PG; 1, 0⟩ = |PG;�1⟩ as follows:
+XG(�2)|X;�1⟩ = |X;�3⟩ = |X; 0, 1⟩,
+(93)
+and
+XG(�2)|PG;�1⟩ = ω9(−�2)|PG;�1⟩.
+(94)
+XG(�5) acts on the state |PL; 1, 0⟩ as follows:
+XG(�2)|PL; 1, 0⟩ = XG(�2)
+�
+j1,j0
+|X; j0, j1⟩⟨X; j0, j1|PL; 1, 0⟩
+= 1
+3XG(�2)
+�
+j1,j0
+|X; �
+j0 + 3j1⟩ω3(j0) = 1
+3
+�
+j1,j0
+ω3(j0)|X;�2 + �
+j0 + 3j1⟩
+= 1
+3{ω3(−1)[|X; �
+−2⟩ + |X;�1⟩ + |X;�4⟩] + [|X; �
+−1⟩ + |X;�2⟩ + |X;�5⟩] + ω3(1)[|X;�0⟩ + |X;�3⟩ + |X;�6⟩]}.(95)
+Eqs. (90), (91), (92) involve local displacements and should be compared and contrasted to Eqs. (93), (95),
+(94) correspondingly, that involve global displacements.
+
+15
+B.
+Local and global position and momentum operators and time evolution
+We define local position and local momentum operators for the r-component of the system as
+X (r)
+L
+= 1 ⊗ ... ⊗ 1 ⊗ X ⊗ 1 ⊗ ... ⊗ 1;
+r = 0, ..., n − 1;
+P(r)
+L
+= 1 ⊗ ... ⊗ 1 ⊗ P ⊗ 1 ⊗ ... ⊗ 1;
+FLPLF †
+L = −XL.
+(96)
+The X, P have been defined in Eq.(9). We can also define global position and global momentum operators as
+XG = −idn
+2π log[ZG(�1)];
+PG = idn
+2π log[XG(�1)];
+FGPGF †
+G = −XG.
+(97)
+They all are dn × dn matrices which can be interpreted as position and momentum operators. In a multipartite
+system with weak interaction between the various parties, it can be argued that the local variables X (r)
+L , P(r)
+L
+are
+more physical operators and the Hamiltonian should be expressed in terms of them. But in the case of strong
+interactions between the parties, the global variables XG, PG might be better for a holistic simple description
+of the physical Hamiltonian with a good approximation.
+Example V.2. We consider the case d = 3, n = 2 and the quantum state
+|s⟩ =
+1
+√
+84(|X; 1⟩ + 2|X; 0⟩ − 3|X; −1⟩) ⊗ (i|X; 1⟩ + |X; 0⟩ − 2i|X; −1⟩).
+(98)
+We also consider time evolution with the Hamiltonians
+h1 = 1
+2[P2
+G + X 2
+G];
+h2 = 1
+2(P2 ⊗ 1) + 1
+2(X 2 ⊗ 1) + 1
+2(1 ⊗ P2) + 1
+2(1 ⊗ X 2) + X ⊗ X.
+(99)
+The first uses the global momentum and position, and the second uses the local momenta and positions. Here
+(in the position basis)
+X = |X; 1⟩⟨X; 1| − |X; −1⟩⟨X; −1|;
+P = −F †XF
+(100)
+Using both notations XG and PG are
+XG =
+1
+�
+a=−1
+1
+�
+b=−1
+(a + 3b)|X; a, b⟩⟨X; a, b| =
+1
+�
+a=−1
+1
+�
+b=−1
+(a + 3b)|X; �
+a + 3b⟩⟨X; �
+a + 3b|
+PG = −F †
+GXGFG.
+(101)
+At time t = 1 the state becomes
+exp(ith1)|s⟩ =
+1
+�
+a=−1
+1
+�
+b=−1
+λ1(a, b)|X; a, b⟩;
+exp(ith2)|s⟩ =
+1
+�
+a=−1
+1
+�
+b=−1
+λ2(a, b)|X; a, b⟩
+(102)
+where
+λ1(−1, −1) = −0.2337 − 0.3556i; λ1(−1, 0) = 0.0259 − 0.1315i; λ1(−1, 1) = 0.3254 + 0.4898i;
+λ1(0, −1) = 0.1160 + 0.0430i; λ1(0, 0) = 0.2836 − 0.0628i; λ1(0, 1) = −0.3090 + 0.1367i;
+λ1(1, −1) = 0.0671 + 0.0943i; λ1(1, 0) = −0.3539 − 0.0845i; λ1(1, 1) = 0.3014 − 0.0675i;
+(103)
+and
+λ2(−1, −1) = −0.2911 − 0.7197i; λ2(−1, 0) = −0.1491 − 0.2668i; λ2(−1, 1) = 0.1502 − 0.0831i;
+λ2(0, −1) = 0.3700 − 0.0854i; λ2(0, 0) = 0.1218 + 0.1319i; λ2(0, 1) = −0.2208 − 0.0247i;
+λ2(1, −1) = 0.0872 − 0.1268i; λ2(1, 0) = −0.0032 + 0.0910i; λ2(1, 1) = −0.0340 − 0.1243i.
+(104)
+
+16
+C.
+Global Wigner and global Weyl functions in Z(dn) × Z(dn)
+If ρ is a density matrix, we define the global Wigner function WG(�γ, �δ|ρ) and the global Weyl function
+�
+WG(�α, �β|ρ) as:
+WG(�γ, �δ|ρ) = Tr[ρPG(�γ, �δ)];
+�
+WG(�α, �β|ρ) = Tr[ρDG(�α, �β)].
+(105)
+From Eq.(86) it follows that they are related to each other through the Fourier transform:
+WG(�γ, �δ|ρ) = 1
+dn
+�
+�α,�β
+�
+WG(�α, �β|ρ)ωdn(�β�γ − �α�δ);
+�
+WG(�α, �β|ρ) = 1
+dn
+�
+�γ,�δ
+WG(�γ, �δ|ρ)ωdn(−�β�γ + �α�δ).
+(106)
+The following marginal properties of the Wigner function follow immediately from Eq.(14) (for odd values of
+the dimension d):
+1
+dn
+�
+{γr}
+WL({γr, δr}|ρ) = 1
+dn
+�
+�γ
+WG(�γ, �δ|ρ) = ⟨X; δ0, ..., δn−1|ρ|X; δ0, ..., δn−1⟩ = ⟨X; �δ|ρ|X; �δ⟩;
+1
+dn
+�
+{δr}
+WL({γr, δr}|ρ) = ⟨PL; γ0, ..., γn−1|ρ|PL; γ0, ..., γn−1⟩;
+1
+dn
+�
+�δ
+WG(�γ, �δ|ρ) = ⟨PG; �γ|ρ|PG; �γ⟩;
+1
+dn
+�
+{γr,δr}
+WL({γr, δr}|ρ) = 1
+dn
+�
+�γ,�δ
+WG(�γ, �δ|ρ) = 1.
+(107)
+We have already emphasized that in both the ‘local formalism’ and the ‘global formalism’ the position states are
+the same and the momentum states are different. Consequently ⟨PL; γ0, ..., γn−1|ρ|PL; γ0, ..., γn−1⟩ is different
+from ⟨PG; �γ|ρ|PG; �γ⟩ and the marginal properties in the second and third of these equations are different.
+D.
+The difference between the local and global Wigner functions
+We first consider states for which the local Wigner function is the same as the global Wigner function.
+Proposition V.3. We consider the following separable density matrix that contains only diagonal elements
+with respect to the basis of position states:
+σ =
+�
+�j
+p(�j)|X;�j⟩⟨X;�j|;
+�
+�j
+p(�j) = 1.
+(108)
+Here the p(�j) are probabilities. In this case the local and global Wigner functions are equal to each other, they
+are non-negative and they do not depend on �α:
+WG(�α, �β|σ) = WL({αr, βr}|σ) = p(�β).
+(109)
+Proof. For the position states
+σ0(�j) = |X;�j⟩⟨X;�j| = |X; j0, ..., jn−1⟩⟨X; j0, ..., jn−1|,
+(110)
+
+17
+we get
+WG(�α, �β|σ0(�j)) = WL({αr, βr}|σ0(�j)) = δ(�j, �β).
+(111)
+Indeed
+WG(�α, �β|σ0(�j)) = Tr
+�
+PG(�α, �β)|X;�j⟩⟨X;�j|
+�
+= Tr
+�
+F 2
+G[DG(�α, �β)]†|X;�j⟩⟨X;�j|DG(�α, �β)
+�
+= Tr
+�
+F 2
+G|X;�j − �β⟩⟨X;�j − �β|
+�
+= δ(�j, �β),
+(112)
+and also
+WL({αr, βr}|σ0(�j)) = Tr[PL({αr, βr})|X; j0, ..., jn−1⟩⟨X; j0, ..., jn−1|]
+= Tr
+�
+F 2
+L[DL({αr, βr})]†|X; j0, ..., jn−1⟩⟨X; j0, ..., jn−1|DL({αr, βr})
+�
+= Tr
+�
+F 2
+L|X; j0 − β0, ..., jn−1 − βn−1⟩⟨X; j0 − β0, ..., jn−1 − βn−1|
+�
+= δ(j0, β0)...δ(jn−1, βn−1).
+(113)
+This proves Eq.(111). Then
+WG(�α, �β|σ) =
+�
+�j
+p(�j)WG(�α, �β|σ0(�j)) =
+�
+�j
+p(�j)δ(�j, �β) = p(�β).
+(114)
+and also
+WL({αr, βr}|σ) =
+�
+j0,...,jn−1
+p(j0, ..., jn−1)WL({αr, βr}|σ0(�j))
+=
+�
+j0,...,jn−1
+p(j0, ..., jn−1)δ(j0, β0)...δ(jn−1, βn−1) = p(�β).
+(115)
+An arbitrary density matrix ρ can be written in the basis of position states, as the sum of a separable density
+matrix σ(ρ) that contains the dn diagonal elements (as in Eq.(108)), and a Hermitian matrix τ(ρ) with trace
+zero that contains the d2n − dn off-diagonal elements:
+ρ = σ(ρ) + τ(ρ);
+Tr[τ(ρ)] = 0.
+(116)
+τ(ρ) is not a density matrix but using Eqs(32), (105) we can define ‘Wigner-like’ functions for it. Then the
+Wigner function is written as a sum of two terms that correspond to the diagonal and off-diagonal part:
+WL({αr, βr}|ρ) = WL({αr, βr}|σ(ρ)) + AL({αr, βr}|τ(ρ));
+AL({αr, βr}|τ(ρ)) = Tr[τ(ρ)PL({αr, βr})];(117)
+and
+WG(�α, �β|ρ) = WG(�α, �β|σ(ρ)) + AG(�α, �β|τ(ρ));
+AG(�α, �β|τ(ρ)) = Tr[τ(ρ)PG(�α, �β)].
+(118)
+Then
+WL({αr, βr}|ρ) − WG(�α, �β|ρ) = AL({αr, βr}|τ(ρ)) − AG(�α, �β|τ(ρ)).
+(119)
+The difference between local and global Wigner functions, is related only to the off-diagonal elements of the
+density matrix (with respect to the position basis).
+
+18
+Proposition V.4. We consider the following separable density matrices
+qL =
+�
+�j
+p(�j)|PL;�j⟩⟨PL;�j|;
+qG =
+�
+�j
+p(�j)|PG;�j⟩⟨PG;�j|;
+F †
+LqLFL = F †
+GqGFG;
+�
+�j
+p(�j) = 1.
+(120)
+Here the p(�j) are probabilities. Then
+WG(�α, �β|qG) = WL({αr, βr}|qL) = p(�α).
+(121)
+Proof. We first consider the density matrices
+qL0(�j) = |PL;�j⟩⟨PL;�j| = |PL; j0, ..., jn−1⟩⟨PL; j0, ..., jn−1|;
+qG0(�j) = |PG;�j⟩⟨PG;�j|.
+(122)
+and prove that
+WG(�α, �β|qG0(�j)) = WL({αr, βr}|qL0(�j)) = δ(�j, �α).
+(123)
+Indeed
+WG(�α, �β|qG0(�j)) = Tr
+�
+PG(�α, �β)|PG;�j⟩⟨PG;�j|
+�
+= Tr
+�
+F 2
+G[DG(�α, �β)]†|PG;�j⟩⟨PG;�j|DG(�α, �β)
+�
+= Tr
+�
+F 2
+G|PG;�j − �α⟩⟨PG;�j − �α|
+�
+= δ(�j, �α),
+(124)
+and
+WL({αr, βr}|qL0(�j)) = Tr[PL({αr, βr})|PL; j0, ..., jn−1⟩⟨PL; j0, ..., jn−1|]
+= Tr
+�
+F 2
+L[DL({αr, βr})]†|PL; j0, ..., jn−1⟩⟨PL; j0, ..., jn−1|DL({αr, βr})
+�
+= Tr
+�
+F 2
+L|PL; j0 − α0, ..., jn−1 − αn−1⟩⟨PL; j0 − α0, ..., jn−1 − αn−1|
+�
+= δ(j0, α0)...δ(jn−1, αn−1).
+(125)
+This proves Eq.(123). Then
+WG(�α, �β|qG) =
+�
+�j
+p(�j)WG(�α, �β|qG0(�j)) =
+�
+�j
+p(�j)δ(�j, �α) = p(�α).
+(126)
+and also
+WL({αr, βr}|qL) =
+�
+j0,...,jn−1
+p(�j)WL({αr, βr}|qL0(�j))
+=
+�
+j0,...,jn−1
+p(j0, ..., jn−1)δ(j0, α0)...δ(jn−1, αn−1) = p(�α).
+(127)
+Example V.5. We consider the density matrices
+ρL = |PL;�j⟩⟨PL;�j|;
+ρG = |PG;�j⟩⟨PG;�j|;
+ρ0 = 1
+dn 1dn.
+(128)
+Then
+σ(ρL) = σ(ρG) = σ(ρ0) = ρ0 = 1
+dn 1dn;
+τ(ρL) = |PL;�j⟩⟨PL;�j| − 1
+dn 1dn;
+τ(ρG) = |PL;�j⟩⟨PL;�j| − 1
+dn 1dn;
+τ(ρ0) = 0.
+(129)
+
+19
+In this case
+WG(�α, �β|ρ0) = WL({αr, βr}|ρ0) = 1
+dn .
+(130)
+From proposition V.4 it follows that
+WG(�α, �β|ρG) = WL(�α, �β|ρL) = δ(�j, �α).
+(131)
+We next calculate numerically the WL(�α, �β|ρG), WG(�α, �β|ρL), for an example. We consider the case d = 3 and
+n = 2, and the density matrices ρL, ρG with �j = 4 (which is an element of Z(9)). Results for the WL(�α, �β|ρG),
+WG(�α, �β|ρL), given in tables I, II correspondingly.
+E.
+The RL, RG matrices: indicators of classical and quantum correlations
+In this section we compare quantities for ρ with the corresponding quantities for R(ρ) (in Eq.(16)).
+Definition V.6. If ρ is a density matrix, RL, �RL are dn × dn matrices with elements
+RL({γr, δr}|ρ) = WL({γr, δr}|ρ) − WL({γr, δr}|R(ρ));
+�RL({αr, βr}|ρ) = �
+WL({αr, βr}|ρ) − �
+WL({αr, βr}|R(ρ)).
+(132)
+Also RG, �RG are dn × dn matrices with elements
+RG((�γ, �δ|ρ) = WG(�γ, �δ|ρ) − WG(�γ, �δ|R(ρ));
+�RG(�α, �β|ρ) = �
+WG(�α, �β|ρ) − �
+WG(�α, �β|R(ρ)).
+(133)
+Proposition V.7.
+(1) For factorisable density matrices RL = �RL = RG = �RG = 0.
+(2) The RL and �RL are related through a local Fourier transform:
+RL({γr, δr}|ρ) = 1
+dn
+�
+{αr,βr}
+�RL({αr, βr}|ρ)ωd
+�n−1
+�
+r=0
+(βrγr − αrδr)
+�
+;
+�RL({αr, βr}|ρ) = 1
+dn
+�
+{γr,δr}
+RL({γr, δr}|ρ)ωd
+�n−1
+�
+r=0
+(−βrγr + αrδr)
+�
+.
+(134)
+(3) The RG and �RG are related through a global Fourier transform:
+RG(�γ, �δ|ρ) = 1
+dn
+�
+�α,�β
+�RG(�α, �β|ρ)ωdn(�β�γ − �α�δ);
+�RG(�α, �β|ρ) = 1
+dn
+�
+�γ,�δ
+RG(�γ, �δ|ρ)ωdn(−�β�γ + �α�δ).
+(135)
+
+20
+(4) The following are marginal properties:
+1
+dn
+�
+{γr}
+RL({γr, δr}|ρ) = 1
+dn
+�
+�γ
+RG(�γ, �δ|ρ) = C(X; δ0, ..., δn−1) = C(X; �δ);
+1
+dn
+�
+{δr}
+RL({γr, δr}|ρ) = C(PL; γ0, ..., γn−1);
+1
+dn
+�
+�δ
+RG(�γ, �δ|ρ) = E(PG; �γ);
+1
+dn
+�
+{γr,δr}
+RL({γr, δr}|ρ) = 1
+dn
+�
+�γ,�δ
+RG(�γ, �δ|ρ) = 0.
+(136)
+Proof.
+(1) For factorisable density matrices R(ρ) = ρ and then RL = �RL = RG = �RG = 0.
+(2) We prove this using Eq.(31) with both ρ and R(ρ).
+(3) We prove this using Eq.(78) with both ρ and R(ρ).
+(4) We prove this using Eq.(107) with both ρ and R(ρ).
+The matrices Rρ and �Rρ indicate the existence of both classical and quantum correlations.
+VI.
+EXAMPLES
+In the examples below we take d = 3 and n = 2. In this case the global Fourier transform is unitarily
+inequivalent to the local Fourier transform. We work in the ‘periods ’ of Eq.(34).
+We consider the density matrix
+ρ = |s⟩⟨s|;
+|s⟩ =
+1
+√
+3|X; 0, 1⟩ + 1
+2|X; 1, −1⟩ +
+�
+5
+12|X; −1, 0⟩.
+(137)
+The state described by ρ is entangled. In this case the reduced density matrices are
+˘ρ0 = 1
+3|X; 0⟩⟨X; 0| + 1
+4|X; 1⟩⟨X; 1| + 5
+12|X; −1⟩⟨X; −1|;
+˘ρ1 = 5
+12|X; 0⟩⟨X; 0| + 1
+3|X; 1⟩⟨X; 1| + 1
+4|X; −1⟩⟨X; −1|.
+(138)
+In tables III,IV,V and VI we present the local Wigner function WL(γ0, γ1; δ0, δ1), the local Weyl function
+�
+WL(α0, α1; β0; β1), and the matrices RL(γ0, γ1; δ0, δ1) and �RL(α0, α1; β0; β1) for the density matrix ρ in Eq.(137).
+The correlations in Eqs.(22),(23) are
+Cρ(X; δ0, δ1) =
+
+
+−0.1042
+0.2431
+−0.1389
+−0.0833 −0.1389
+0.2222
+0.1875
+−0.1042 −0.0833
+
+ ;
+Cρ(PL; γ0, γ1) =
+
+
+−0.1093 −0.1093
+0.2187
+−0.1093
+0.2187
+−0.1093
+0.2187
+−0.1093 −0.1093
+
+ .
+(139)
+We easily confirm that Eqs(107) hold for the local Wigner and Weyl function.
+For the global formalism in Z(9) we rewrite ρ as
+ρ = |s⟩⟨s|;
+|s⟩ =
+1
+√
+3|X;�3⟩ + 1
+2|X; �
+−2⟩ +
+�
+5
+12|X; �
+−1⟩.
+(140)
+
+21
+In tables VII,VIII, IX and X we present the global Wigner function WG(�γ, �δ|ρ), the global Weyl function
+�
+WG(�α, �β|ρ) and the matrices RG(�γ, �δ) and �
+RG(�α, �β) for the density matrix ρ in Eq.(137).
+The correlations in Eq.(61) are
+Cρ(X; �δ) = Cρ(X; δ0, δ1);
+Eρ(PG; �γ) =
+�−0.0419 −0.1093 0.1250 −0.0832 0.2187 −0.0832 0.1250 −0.1093 −0.0419�T . (141)
+We easily confirm that Eqs(107) hold for the global Wigner and Weyl function.
+In general there is no simple relation that links the local with the global quantities. We see this by comparing
+the expectation values of the local observables X ⊗ 1, 1 ⊗ X, X ⊗ X, P ⊗ 1, 1 ⊗ P, P ⊗ P for the density
+matrix ρ in Eq.(137), with the expectation values of the global observables XG, PG for the same density matrix
+(written in the ‘global language’ in Eq.(140)):
+Tr[ρ(X ⊗ 1)] = −0.1667;
+Tr[ρ(1 ⊗ X)] = 0.0833;
+Tr[ρ(X ⊗ X)] = −0.25;
+Tr(ρXG) = 0.0833
+Tr[ρ(P ⊗ 1)] = 0;
+Tr[ρ(1 ⊗ P)] = 0;
+Tr[ρ(P ⊗ P)] = −0.6561;
+Tr(ρPG) = 0.
+(142)
+The X, P, XG, PG, have been given in Eqs(100), (101).
+The results for the local observables are different from the global observables. For strongly correlated systems
+global quantities might be physically more relevant.
+We note that an observable can be written in both the local and global formalism (using the map in Eq.(35)).
+For example, for the above system we consider the observable (Hermitian operator)
+O = a|X; 0, 1⟩⟨X; 0, 1| + b|X; 1, −1⟩⟨X; 1, −1| + c|X; −1, 0⟩⟨X; −1, 0|
++ d|X; 0, 1⟩⟨X; 1, −1| + d∗|X; 1, −1⟩⟨X; 0, 1|;
+a, b, c ∈ R;
+d ∈ C,
+(143)
+which can also be written as
+O = a|X;�3⟩⟨X;�3| + b|X; �
+−2⟩⟨X; �
+−2| + c|X; �
+−1⟩⟨X; �
+−1|
++ d|X;�3⟩⟨X; �
+−2| + d∗|X; �
+−2⟩⟨X;�3|.
+(144)
+Important physical quantities like the position can be defined locally like X ⊗ 1, 1 ⊗ X, X ⊗ X, or globally as
+XG (defined in Eq.(97)). The same is true for local and global momenta. For strongly correlated systems the
+identity of each component becomes weak, and global quantities might be physically more appropriate for the
+description of these systems.
+VII.
+DISCUSSION
+In this paper we introduced local and global Fourier transforms and related phase space methods for multi-
+partite systems. The multipartite system consists of n components, each of which is described with variables
+in Z(d) and with a d-dimensional Hilbert space H(d). In the global formalism we take a holistic view of the
+system and describe it with variables in [Z(dn)] and the dn-dimensional Hilbert space H. Even if the various
+components of the system are located far from each other, in the case of strong interactions and strong correla-
+tions between them they might loose their individual identity. In this case a holistic approach that uses global
+quantities, might be more appropriate.
+In the local formalism the phase space is [Z(d) × Z(d)]n, and in the global formalism [Z(dn)] × [Z(dn)]. We
+have explained that although the map in Eq.(35) is bijective, the ring [Z(d)]n is not isomorphic to the ring
+[Z(dn)] (because of Eq.(36)). The heart of the formalism is the local and global Fourier transforms. We have
+shown that for some values of d, n they are unitarily inequivalent to each other (proposition IV.4).
+We have compared and contrasted the local phase space formalism with the global phase space formalism.
+Examples of this are:
+
+22
+• Some of the local momentum states are the same as the global momentum states (proposition IV.3).
+• Density matrices which have only diagonal elements with respect to the position basis, have the same local
+and global Wigner function (proposition V.3). The difference between local and global Wigner functions,
+is contained entirely in the off-diagonal elements.
+• We have calculated the time evolution in terms of both local variables and also global variables (section
+V B)
+• Classical and quantum correlations have been described in the local formalism with the matrices RL, �RL
+and in the global formalism with the matrices RG, �RG.
+The formalism could be used in the general area of Fast Fourier transforms (in a quantum or even classical
+context). For example, a link between the present formalism (in some special cases) and the Cooley-Tukey
+formalism has been discussed in section IV E.
+The work is a contribution to the various approaches for multipartite systems. Unitary equivalence between
+the local and global Fourier transform (Eq.(64)), implies that the distinction between the concept of a multi-
+partite system and that of a single system is weak. Unitary inequivalence (Eq.(65) ) implies that the concept
+of a multipartite system is fundamentally different from that of a single quantum system.
+Conflict of interest and data availability statement
+We have no conflicts of interest to disclose.
+No data were used in this paper, and therefore data availability is not applicable.
+References
+[1] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, ‘Quantum entanglement’, Rev. Mod. Phys. 81, 865 (2009)
+[2] A. Vourdas, ‘Finite and profinite quantum systems’ (Springer, Berlin, 2017)
+[3] T. Durt, B.G. Englert, I. Bengtsson, K. Zyczkowski, ‘On mutually unbiased bases’, Int. J. Quantum Comput. 8, 535
+(2010)
+[4] A. Vourdas, ‘Multipartite quantum systems: an approach based on Markov matrices and the Gini index’, J. Phys
+A54, 185201 (2021)
+[5] J.H. McClellan, C.M. Rader, ‘Number theory in digital signal processing’ (Prentice Hall, New Jersey, 1979)
+[6] R.E. Blahut ‘Fast algorithms for digital signal processing’ Addison-Wesley, Reading Mass, 1985)
+[7] D.F. Elliott, K.R. Rao, ‘Fast transforms’ (Academic Press, London, 1982)
+[8] M. Horibe, A. Takami, T. Hashimoto, A. Hayashi, ‘Existence of the Wigner function with correct marginal distri-
+butions along tilted lines on a lattice’, Phys. Rev. A65, 032105 (2002)
+[9] T. Durt, ‘About mutually unbiased bases in even and odd prime power dimensions’, J. Phys. A38, 5267 (2005)
+[10] J. Zak, ‘Doubling feature of the Wigner function: finite phase space’, J. Phys. A44, 345305 (2011)
+[11] A. Terras ‘Fourier analysis on finite groups and applications (Cambridge Univ. Press, Cambridge, 1999)
+[12] I.J. Good, ‘The relationship between two fast Fourier transforms’, IEEE Transactions on computers C-20, 310 (1971)
+[13] A. Vourdas, ‘Factorisation in finite quantum systems’, J. Phys. A36, 5645 (2003)
+[14] T.G. Gerasimova, ‘Unitary similarity to a normal matrix’, Linear Algebra Appl. 436, 3777 (2012)
+[15] H. Shapiro, ‘A survey of canonical forms and invariants for unitary similarity’, Linear Algebra Appl. 147, 101 (1991)
+
+23
+{β0,β1}
+{−1, −1} {0, −1}
+{1, −1}
+{−1, 0}
+{0, 0}
+{1, 0}
+{−1, 1}
+{0, 1}
+{1, 1}
+{−1, −1}
+0
+0
+0
+0
+0
+0
+0
+0
+0
+{0, −1}
+0
+0
+0
+0
+0
+0
+0
+0
+0
+{1, −1}
+0
+0
+0
+0
+0
+0
+0
+0
+0
+{−1, 0}
+0
+0
+0
+0
+0
+0
+0
+0
+0
+{α0,α1}
+{0, 0}
+0
+0
+0
+0
+0
+0
+0
+0
+0
+{1, 0}
+0
+0
+0
+0
+0
+0
+0
+0
+0
+{−1, 1}
+0.4491
+−0.2931
+0.4491
+0.4491
+−0.2931
+0.4491
+0.4491
+−0.2931
+0.4491
+{0, 1}
+−0.2931
+0.844
+−0.2931 −0.2931
+0.844
+−0.2931 −0.2931
+0.844
+−0.2931
+{1, 1}
+0.844
+0.4491
+0.844
+0.844
+0.4491
+0.844
+0.844
+0.4491
+0.844
+TABLE I: The local Wigner function WL({α0, α1; β0, β1}|ρG) for the density matrix ρG in Eq.(128) with �j = 4 and
+d = 3, n = 2.
+�β = (β0, β1)
+�
+−4 = (−1, −1) �
+−3 = (0, −1) �
+−2 = (1, −1) �
+−1 = (1, 0)
+�0 = (0, 0)
+�1 = (1, 0) �2 = (−1, 1) �3 = (0, 1) �4 = (1, 1)
+�
+−4 = (−1, −1)
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�
+−3 = (0, −1)
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�
+−2 = (1, −1)
+−0.2931
+0.844
+−0.2931
+−0.2931
+0.844
+−0.2931
+−0.2931
+0.844
+−0.2931
+�
+−1 = (−1, 0)
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�α = (α0, α1)
+�0 = (0, 0)
+0
+0
+0
+0
+�0
+0
+0
+0
+0
+�1 = (1, 0)
+0.4491
+−0.2931
+0.4491
+0.4491
+−0.2931
+0.4491
+0.4491
+−0.2931
+0.4491
+�2 = (−1, 1)
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�3 = (0, 1)
+0
+0
+0
+0
+0
+0
+0
+0
+0
+�4 = (1, 1)
+0.844
+0.4491
+0.844
+0.844
+0.4491
+0.844
+0.844
+0.4491
+0.844
+TABLE II: The global Wigner function WG(�α, �β|ρL) for the density matrix ρL in Eq.(128), with �j = 4 and d = 3, n = 2.
+{δ0,δ1}
+{−1, −1} {0, −1} {1, −1} {−1, 0} {0, 0} {1, 0} {−1, 1} {0, 1} {1, 1}
+{−1, −1}
+0
+0
+−0.1227
+0.128
+0
+0
+0
+0.0106
+0
+{0, −1}
+0
+0
+−0.1227
+0.128
+0
+0
+0
+0.0106
+0
+{1, −1}
+0
+0
+0.9954
+0.994
+0
+0
+0
+0.9788
+0
+{−1, 0}
+0
+0
+−0.1227
+0.128
+0
+0
+0
+0.0106
+0
+{γ0,γ1}
+{0, 0}
+0
+0
+0.9954
+0.994
+0
+0
+0
+0.9788
+0
+{1, 0}
+0
+0
+−0.1227
+0.128
+0
+0
+0
+0.0106
+0
+{−1, 1}
+0
+0
+0.9954
+0.994
+0
+0
+0
+0.9788
+0
+{0, 1}
+0
+0
+−0.1227
+0.128
+0
+0
+0
+0.0106
+0
+{1, 1}
+0
+0
+−0.1227
+0.128
+0
+0
+0
+0.0106
+0
+TABLE III: The local Wigner function WL({γ0, γ1; δ0, δ1}|ρ) for the density matrix ρ in Eq.(137).
+
+24
+{β0,β1}
+{−1, −1}
+{0, −1} {1, −1} {−1, 0}
+{0, 0}
+{1, 0} {−1, 1} {0, 1}
+{1, 1}
+{−1, −1}
+0.067 − 0.0295i
+0
+0
+0
+−0.125 + 0.0722i
+0
+0
+0
+0.067 − 0.0295i
+{0, −1}
+−0.059 + 0.0432i
+0
+0
+0
+0.125 − 0.0722i
+0
+0
+0
+−0.059 + 0.0432i
+{1, −1}
+−0.4921 − 0.8523i
+0
+0
+0
+−0.5 − 0.866i
+0
+0
+0
+−0.4921 − 0.8523i
+{−1, 0}
+−0.0079 − 0.0727i
+0
+0
+0
+0.1443i
+0
+0
+0
+−0.0079 − 0.0727i
+{α0,α1}
+{0, 0}
+0.9841
+0
+0
+0
+1
+0
+0
+0
+0.9841
+{1, 0}
+−0.0079 + 0.0727i
+0
+0
+0
+−0.1443i
+0
+0
+0
+−0.0079 + 0.0727i
+{−1, 1}
+−0.4921 + 0.8523i
+0
+0
+0
+−0.5 + 0.866i
+0
+0
+0
+−0.4921 + 0.8523i
+{0, 1}
+−0.059 − 0.0432i
+0
+0
+0
+0.125 + 0.0722i
+0
+0
+0
+−0.059 − 0.0432i
+{1, 1}
+0.067 + 0.0295i
+0
+0
+0
+−0.125 − 0.0722i
+0
+0
+0
+0.067 + 0.0295i
+TABLE IV: The local Weyl function �
+WL({α0, α1; β0, β1}|ρ) for the density matrix ρ in Eq.(137)
+{δ0,δ1}
+{−1, −1} {0, −1}
+{1, −1}
+{−1, 0}
+{0, 0}
+{1, 0}
+{−1, 1}
+{0, 1}
+{1, 1}
+{−1, −1} −0.1042 −0.0833 −0.1852 −0.0456 −0.1389 −0.1042 −0.1389 −0.1005 −0.0833
+{0, −1}
+−0.1042 −0.0833 −0.1852 −0.0456 −0.1389 −0.1042 −0.1389 −0.1005 −0.0833
+{1, −1}
+−0.1042 −0.0833
+0.9329
+0.8204
+−0.1389 −0.1042 −0.1389
+0.8677
+−0.0833
+{−1, 0}
+−0.1042 −0.0833 −0.1852 −0.0456 −0.1389 −0.1042 −0.1389 −0.1005 −0.0833
+{γ0,γ1}
+{0, 0}
+−0.1042 −0.0833
+0.9329
+0.8204
+−0.1389 −0.1042 −0.1389
+0.8677
+−0.0833
+{1, 0}
+−0.1042 −0.0833 −0.1852 −0.0456 −0.1389 −0.1042 −0.1389 −0.1005 −0.0833
+{−1, 1}
+−0.1042 −0.0833
+0.9329
+0.8204
+−0.1389 −0.1042 −0.1389
+0.8677
+−0.0833
+{0, 1}
+−0.1042 −0.0833 −0.1852 −0.0456 −0.1389 −0.1042 −0.1389 −0.1005 −0.0833
+{1, 1}
+−0.1042 −0.0833 −0.1852 −0.0456 −0.1389 −0.1042 −0.1389 −0.1005 −0.0833
+TABLE V: The matrix RL({γ0, γ1; δ0, δ1}|ρ) for the density matrix ρ in Eq.(137).
+{β0,β1}
+{−1, −1}
+{0, −1} {1, −1} {−1, 0}
+{0, 0}
+{1, 0} {−1, 1} {0, 1}
+{1, 1}
+{−1, −1}
+0.067 − 0.0295i
+0
+0
+0
+−0.1354 + 0.0541i
+0
+0
+0
+0.067 − 0.0295i
+{0, −1}
+−0.059 + 0.0432i
+0
+0
+0
+0
+0
+0
+0
+−0.059 + 0.0432i
+{1, −1}
+−0.4921 − 0.8523i
+0
+0
+0
+−0.4896 − 0.848i
+0
+0
+0
+−0.4921 − 0.8523i
+{−1, 0}
+−0.0079 − 0.0727i
+0
+0
+0
+0
+0
+0
+0
+−0.0079 − 0.0727i
+{α0,α1}
+{0, 0}
+0.9841
+0
+0
+0
+0
+0
+0
+0
+0.9841
+{1, 0}
+−0.0079 + 0.0727i
+0
+0
+0
+0
+0
+0
+0
+−0.0079 + 0.0727i
+{−1, 1}
+−0.4921 + 0.8523i
+0
+0
+0
+−0.4896 + 0.848i
+0
+0
+0
+−0.4921 + 0.8523i
+{0, 1}
+−0.059 − 0.0432i
+0
+0
+0
+0
+0
+0
+0
+−0.059 − 0.0432i
+{1, 1}
+0.067 + 0.0295i
+0
+0
+0
+−0.1354 − 0.0541i
+0
+0
+0
+0.067 + 0.0295i
+TABLE VI: The matrix �
+RL({α0, α1; β0, β1}|ρ) for the density matrix ρ in Eq.(137).
+�δ = (δ0, δ1)
+�
+−4 = (−1, −1) �
+−3 = (0, −1) �
+−2 = (1, −1) �
+−1 = (1, 0)
+�0 = (0, 0)
+�1 = (1, 0) �2 = (−1, 1) �3 = (0, 1) �4 = (1, 1)
+�
+−4 = (−1, −1)
+0.1003
+0
+0.25
+0.4167
+0
+0.1294
+0
+−0.2732
+0
+�
+−3 = (0, −1)
+−0.2887
+0
+0.25
+0.4167
+0
+−0.3727
+0
+0.0106
+0
+�
+−2 = (1, −1)
+0.4423
+0
+0.25
+0.4167
+0
+0.571
+0
+0.4454
+0
+�
+−1 = (−1, 0)
+−0.5425
+0
+0.25
+0.4167
+0
+−0.7004
+0
+0.8278
+0
+�γ = (γ0, γ1)
+�0 = (0, 0)
+0.5774
+0
+0.25
+0.4167
+0
+0.7454
+0
+0.9788
+0
+�1 = (1, 0)
+−0.5425
+0
+0.25
+0.4167
+0
+−0.7004
+0
+0.8278
+0
+�2 = (−1, 1)
+0.4423
+0
+0.25
+0.4167
+0
+0.571
+0
+0.4454
+0
+�3 = (0, 1)
+−0.2887
+0
+0.25
+0.4167
+0
+−0.3727
+0
+0.0106
+0
+�4 = (1, 1)
+0.1003
+0
+0.25
+0.4167
+0
+0.1294
+0
+−0.2732
+0
+TABLE VII: The global Wigner function WG(�γ, �δ|ρ) for the density matrix ρ in Eq.(137).
+
+25
+�β = (β0, β1)
+�
+−4 = (−1, −1)
+�
+−3 = (0, −1) �
+−2 = (1, −1)
+�
+−1 = (1, 0)
+�0 = (0, 0)
+�1 = (1, 0)
+�2 = (−1, 1) �3 = (0, 1)
+�4 = (1, 1)
+�
+−4 = (−1, −1) −0.3001 − 0.4118i
+0
+0
+−0.1614 − 0.2795i −0.3667 − 0.3069i −0.1614 − 0.2795i
+0
+0
+−0.3001 − 0.4118i
+�
+−3 = (0, −1)
+−0.3307 − 0.0727i
+0
+0
+0.3227
+0.1443i
+0.3227
+0
+0
+−0.3307 − 0.0727i
+�
+−2 = (1, −1)
+0.2859 − 0.5526i
+0
+0
+−0.1614 + 0.2795i −0.3292 + 0.7845i −0.1614 + 0.2795i
+0
+0
+0.2859 − 0.5526i
+�
+−1 = (−1, 0)
+0.0142 − 0.1408i
+0
+0
+−0.1614 − 0.2795i
+0.1959 + 0.2254i
+−0.1614 − 0.2795i
+0
+0
+0.0142 − 0.1408i
+�α = (α0, α1)
+�0 = (0, 0)
+0.6614
+0
+0
+0.3227
+1
+0.3227
+0
+0
+0.6614
+�1 = (1, 0)
+0.0142 + 0.1408i
+0
+0
+−0.1614 + 0.2795i
+0.1959 − 0.2254i
+−0.1614 + 0.2795i
+0
+0
+0.0142 + 0.1408i
+�2 = (−1, 1)
+0.2859 + 0.5526i
+0
+0
+−0.1614 − 0.2795i −0.3292 − 0.7845i −0.1614 − 0.2795i
+0
+0
+0.2859 + 0.5526i
+�3 = (0, 1)
+−0.3307 + 0.0727i
+0
+0
+0.3227
+−0.1443i
+0.3227
+0
+0
+−0.3307 + 0.0727i
+�4 = (1, 1)
+−0.3001 + 0.4118i
+0
+0
+−0.1614 + 0.2795i −0.3667 + 0.3069i −0.1614 + 0.2795i
+0
+0
+−0.3001 + 0.4118i
+TABLE VIII: The global Weyl function �
+WG(�α, �β|ρ) for the density matrix ρ in Eq.(137).
+�δ = (δ0, δ1)
+�
+−4 = (−1, −1) �
+−3 = (0, −1) �
+−2 = (1, −1) �
+−1 = (1, 0)
+�0 = (0, 0)
+�1 = (1, 0) �2 = (−1, 1) �3 = (0, 1) �4 = (1, 1)
+�
+−4 = (−1, −1)
+−0.0039
+−0.0833
+0.1875
+0.2431
+−0.1389
+0.0253
+−0.1389
+−0.3843
+−0.0833
+�
+−3 = (0, −1)
+−0.3928
+−0.0833
+0.1875
+0.2431
+−0.1389
+−0.4768
+−0.1389
+−0.1005
+−0.0833
+�
+−2 = (1, −1)
+0.3381
+−0.0833
+0.1875
+0.2431
+−0.1389
+0.4668
+−0.1389
+0.3343
+−0.0833
+�
+−1 = (−1, 0)
+−0.6467
+−0.0833
+0.1875
+0.2431
+−0.1389
+−0.8046
+−0.1389
+0.7167
+−0.0833
+�γ = (γ0, γ1)
+�0 = (0, 0)
+0.4732
+−0.0833
+0.1875
+0.2431
+−0.1389
+0.6412
+−0.1389
+0.8677
+−0.0833
+�1 = (1, 0)
+−0.6467
+−0.0833
+0.1875
+0.2431
+−0.1389
+−0.8046
+−0.1389
+0.7167
+−0.0833
+�2 = (−1, 1)
+0.3381
+−0.0833
+0.1875
+0.2431
+−0.1389
+0.4668
+−0.1389
+0.3343
+−0.0833
+�3 = (0, 1)
+−0.3928
+−0.0833
+0.1875
+0.2431
+−0.1389
+−0.4768
+−0.1389
+−0.1005
+−0.0833
+�4 = (1, 1)
+−0.0039
+−0.0833
+0.1875
+0.2431
+−0.1389
+0.0253
+−0.1389
+−0.3843
+−0.0833
+TABLE IX: The matrix RG(�γ, �δ|ρ) for the density matrix ρ in Eq.(137).
+�β = (β0, β1)
+�
+−4 = (−1, −1)
+�
+−3 = (0, −1) �
+−2 = (1, −1)
+�
+−1 = (1, 0)
+�0 = (0, 0)
+�1 = (1, 0)
+�2 = (−1, 1) �3 = (0, 1)
+�4 = (1, 1)
+�
+−4 = (−1, −1) −0.3001 − 0.4118i
+0
+0
+−0.1614 − 0.2795i −0.3342 − 0.3351i −0.1614 − 0.2795i
+0
+0
+−0.3001 − 0.4118i
+�
+−3 = (0, −1)
+−0.3307 − 0.0727i
+0
+0
+0.3227
+0
+0.3227
+0
+0
+−0.3307 − 0.0727i
+�
+−2 = (1, −1)
+0.2859 − 0.5526i
+0
+0
+−0.1614 + 0.2795i −0.3735 + 0.7316i −0.1614 + 0.2795i
+0
+0
+0.2859 − 0.5526i
+�
+−1 = (−1, 0)
+0.0142 − 0.1408i
+0
+0
+−0.1614 − 0.2795i
+0.0827 + 0.2729i
+−0.1614 − 0.2795i
+0
+0
+0.0142 − 0.1408i
+�α = (α0, α1)
+�0 = (0, 0)
+0.6614
+0
+0
+0.3227
+0
+0.3227
+0
+0
+0.6614
+�1 = (1, 0)
+0.0142 + 0.1408i
+0
+0
+−0.1614 + 0.2795i
+0.0827 − 0.2729i
+−0.1614 + 0.2795i
+0
+0
+0.0142 + 0.1408i
+�2 = (−1, 1)
+0.2859 + 0.5526i
+0
+0
+−0.1614 − 0.2795i −0.3735 − 0.7316i −0.1614 − 0.2795i
+0
+0
+0.2859 + 0.5526i
+�3 = (0, 1)
+−0.3307 + 0.0727i
+0
+0
+0.3227
+0
+0.3227
+0
+0
+−0.3307 + 0.0727i
+�4 = (1, 1)
+−0.3001 + 0.4118i
+0
+0
+−0.1614 + 0.2795i −0.3342 + 0.3351i −0.1614 + 0.2795i
+0
+0
+−0.3001 + 0.4118i
+TABLE X: The matrix �
+RG(�α, �β|ρ) for the density matrix ρ in Eq.(137).
+

39E1T4oBgHgl3EQf6AV3/content/tmp_files/2301.03518v1.pdf.txt

@@ -0,0 +1,389 @@
+arXiv:2301.03518v1  [physics.optics]  9 Jan 2023
+Size Effects in Periodic Metamaterials
+Victor V. Gozhenko
+Institute of Physics, Natl. Acad. of Sciences of Ukraine,
+46 Nauky Ave., Kyiv 03680, Ukraine∗
+1
+
+Abstract
+The optical properties of periodic electromagnetic metamaterials are considered as functions of
+their relative unit cell size d/λ. The reflection R and transmission T coefficients are numerically
+calculated for some realistic metamaterials in a wide range of their relative unit cell size values
+that comprises different operating regimes of the metamaterials. Peculiarities in R and T behavior
+are discussed and the causes of those peculiarities are outlined. The obtained results support the
+opinion on inapplicability of the very homogenization concept to metamaterials whose unit cell size
+is comparable to the incident wavelength, in contrast to some previously published results.
+I.
+INTRODUCTION
+Most of the electromagnetic metamaterials are periodic structures, and their unit cells
+consist of artificial inclusions designed to get a specific electromagnetic response (e.g., neg-
+ative refraction or selective reflectivity) of the metamaterial sample as a whole. Periodic
+metamaterial can be treated as a continuous and homogeneous medium if its unit cell size
+Figure 1. The concept of metamaterials homogenization. A periodic metamaterial with unit cells
+of size d is lit by an incident electromagnetic wave whose wavelength is λ. If d ≪ λ, then the
+incident wave cannot “feel” the metamaterial inhomogenities, and metamaterial behave itself like
+a continuous and homogeneous medium, whose parameters are εeff, µeff, neff. Representation of a
+metamaterial by the corresponding homogeneous medium is correct if the optical properties (e.g.,
+reflectance and transmittance) of the metamaterial and the medium are the same.
+∗ [email protected]
+2
+
+reflected
+transmitted
+Eeff
+Ueff
+neff
+incidentd (the lattice constant) is much smaller then its operating wavelength λ, d ≪ λ. In such a
+case, the metamaterial can be characterized by its effective parameters—the effective per-
+mittivity εeff, permeability µeff, and index of refraction (the refractive index) neff = √εµ, see
+Fig. 1.
+Calculation of the effective parameters values for a given metamaterial (with given shape,
+size, and material of its inclusions, as well as the size and geometry of its unit cell) is im-
+portant, for example, for predicting the optical properties the metamaterial will reveal in
+experiments and applications, and is based on calculating the local electric and magnetic
+fields within the unit cell and proper averaging of those fields over the cell. Sometimes—in
+case of simple inclusions—it can be done analytically; in general case, numerical computa-
+tions are required.
+To facilitate calculation of the effective parameters of periodic metamaterials, a number
+of homogenization theories and methods were proposed (see, e.g., Refs. 1–6). Those methods
+differ from each other by, particularly, the way they calculate the average values of the local
+fields in the metamaterials.
+In metamaterials applications, condition d ≪ λ (say, d = 0.01λ) is not always met. For
+example, optical applications (where λ ≈ 400 . . . 800 nm) implies that the unit cell size
+should be of the order of 50 nm or less. However, metamaterials with d = 200 . . . 300 nm
+(i.e., d/λ ≈ 0.25 . . . 0.5) are often used there because the less the unit cell size, the more
+expensive manufacturing process of the metamaterial sample. On the other hand, some
+applications—most notably the negative index of refraction—require for a metamaterial to
+work in the resonant regime (where the negative n is achieveable) meaning d/λ ≈ 0.5 . . . 1.0.
+Therefore, some authors tried to elaborate homogenization methods suitable for an extended
+range of d/λ values, and not only for small d. Some of them believe that their methods are
+valid for metamaterials with substantial or even arbitrary unit cell size (e.g., Refs. 1, 3, and
+6).
+Strictly speaking, any homogenization method can give plausible results for metamate-
+rials working in the long wavelength (quasistatic) regime only, where the condition d ≪ λ
+is satisfied. In the opposite case of short (relative to the unit cell size) waves, d ≫ λ, the
+very homogenization concept should fail, and metamaterials cannot be treated as homoge-
+neous media. In this case, propagation of incident waves through a metamaterial obeys the
+geometrical optics laws, and reflection of an incident wave from the metamaterial inclusions
+3
+
+plays a crucial role. Last, in the intermediate regime, where d ≈ λ, homogenization meth-
+ods should not work since they do not account for the diffraction effects (e.g., the Bragg’s
+reflection) which are significant in this case.
+Earlier [4], it was shown that different homogenization methods give more and more di-
+verging results as the relative unit cell size of metamaterials increases from zero to approx-
+imately d/λ = 0.4. At larger d/λ values, calculations of homogenized effective parameters
+of a metamaterial can still formally be performed, but those parameters cannot describe
+correctly the optical properties of the metamaterial.
+In the present paper, the optical properties of periodic metamaterials are considered in a
+wider range of their relative unit cell sizes d/λ, and the effects of the cell size on the optical
+behaviour of the matematerials is discussed in more details.
+II.
+BASIC FORMULAE
+We are interested in calculating the observable quantities—transmittance T and re-
+flectance R of a metamaterial—which are dimensionless coefficients defined as
+T = Itr
+I0
+,
+R = Iref
+I0
+,
+where I0, Itr, Iref are the intensities of incident, transmitted through the metamaterial, and
+reflected waves. The intensities are the energy flux densities of the corresponding waves and
+can be calculated as time-averaged values of the Poynting vector S of those waves (indices
+are omitted below for simplicity):
+I = ⟨S⟩ = 1
+τ
+t+τ
+�
+t
+S(t′)dt′
+In case of monochromatic incident plane wave
+E = E0e[i(k·r−ωt)],
+(1)
+H = H0e[i(k·r−ωt)]
+(2)
+all the waves involved are also monochromatic and their intensities can be calculated from
+I = 1
+2Re(E × H∗),
+4
+
+where the asterisk denotes the complex conjugation.
+From the energy conservation law
+applied to the interaction of electromagnetic waves with a lossy medium, it follows
+T + R + A = 1,
+where A is the absorptance (the absorption coefficient) of the medium which define the rate
+of electromagnetic energy absorption inside it. If T and R values for a medium are known
+(i.e., are experimentally measured or theoretically calculated), its absorptance can be found
+as
+A = 1 − T − R.
+III.
+NUMERICAL RESULTS AND DISCUSSION
+Numerical simulations are carried out for metamaterials with cubic lattice and consisted of
+inclusions of various shapes that are often used in metamaterial science and applications—
+spheres, rods, Split-Ring Resonators (SRRs), and Ω-like inclusions (“omegas”).
+Optical
+reflectance R and transmittance T of the simulated metamaterials are calculated in a wide
+range of their relative unit cell size d/λ at several values of the incidence angle θ. Calculations
+of R and transmittance T are complemented with the local E field distributions across
+the unit cells in different regimes the metamaterials operate in. All the calculations are
+performed with COMSOL Multiphysics software. Presented below are exemplary calculation
+results.
+Schematics of the unit cells of the simulated metamaterials are shown in Fig. 2. The
+materials are primarily infinite monolayers of thickness d and that in Fig. 2d is a triple layer
+of thickness 3d. The unit cell size d = 500 nm is the same for all the materials and remains
+unchangeable in all the calculations. Variations in the relative unit cell size d/λ are made
+by changing the wavelength λ of the incident wave.
+From Fig. 3 one can see that all the inclusions give a prominent response to the incident
+wave even at normal incidence, the response of spherical inclusions being dipole-like (see
+panel (a), the field distribution in the middle plane), while those of “SRRs with rods” system
+and “omegas” are more tricky.
+The incidence angle θ also affects the electromagnetic field distribution inside metama-
+terials: even for simple spherical inclusions in the quasistatic regime (λ = 10d) the electric
+5
+
+Figure 2. Unit cells of the simulated metamaterials: (a) a monolayer of spherical particles; (b) a
+monolayer of omegas; (c) a monolayer of SRRs and rods; (d) a triple layer of spherical particles.
+All the inclusions are made of gold, and the size of the individual unit cells is d = 500 nm.
+fields in the unit cell differ substantially at normal and oblique incidence, see Fig. 4.
+Shown in Figs. 5–7 are the transmission and reflection spectra numerically calculated for
+the metamaterials depicted in Fig. 2a,c,d.
+According to Fig. 5, at normal incidence of a long enough wave with λ = 3000 nm (which
+is six times the unit cell size d) onto the monolayer of golden spheres, the metamaterial
+behaves itself as a transparent sheet: its reflection coefficient R in this regime is close to
+zero, and its transmission coefficent T is near unity. If λ decreases and approaches the unit
+cell size d, the metamaterial gradually looses its transparency and becomes more and more
+reflective. The reflection coefficient has its maximum R = 0.45 at λ = 1.2d = 600 nm,
+and the transmission coefficient is minimal (T = 0.3) at this point. Further decrease of λ
+results in monotonic increase of the material transparency (with the peak value T ≈ 0.77 at
+λ = d = 500 nm) and monotonic decrease of its reflectance up to R ≈ 0. The radical change
+in the behavior of R and T at λ ≤ 600 nm (when d/λ ≥ 0.83) is probably due to the onset
+of the diffraction and interference effects in the periodic metamaterial.
+Note that the sum T + R becomes distinctly less then unity at λ ≤ 650 nm, or d/λ ≥
+0.77. It means that light absorption by the metamaterial is substantial in this case. The
+minimum value of the absorption coefficient A = 1 − T + R ≈ 0.67 is observed at λ = 550
+nm, which implies that about one third of the incident energy flux is absorbed by the
+metamaterial inclusions.
+The absorption can be ascribed to electric currents induced in
+individual inclusions (golden spheres) by the incident wave as well scattered waves from
+neighboring particles.
+Analogous behavior is observed at oblique incidence (see the respective curves in Fig. 5
+6
+
+(a)
+(b)
+(c)
+(d)Figure 3. Local distributions of the absolute value of the electric field E inside the unit cells of the
+metamaterials shown in Fig. 2a–c. Left column, the field on the unit cell boundaries; right column,
+the field in the middle plane of the cells. λ = 1000 nm, normal incidence (θ = 0). Directions of
+E and H vectors of the incident wave are depicted in panel (a). Distribution of |E| allows one to
+easily determine those places where the electric energy is concentrated.
+for the case of θ = 45◦): with decreasing λ, the region of insufficient changes in R, T passes
+(starting from λ ≈ 1.8d = 900 nm) to the region of their abrupt changes and oscillating
+behavior. Note, however, that the metamaterial in this case is semi-transparent even at
+large λ: R + T ≈ 0.62 at λ = 3000 nm.
+The plots in Fig. 6 refer to the monolayer of golden inclusions “SRR plus rod” and have
+7
+
+(a)
+X107
+X108
+1.4
+1.3
+1.2
+6.5
+1.1
+6
+1
+E
+0.9
+5.5
+0.8
+H
+0.7
+4.5
+0.6
+0.5
+4
+0.4
+(q)
+X108
+X108
+1
+0.9
+0.8
+0.7
+0.6
+9
+0.8
+0.5
+0.7
+0.4
+0.6
+0.3
+0.5
+0.2
+0.4
+(c)
+X108
+X108
+2.4
+0.9
+2.2
+0.8
+2
+0.7
+1.8
+0.6
+1.6
+0.5
+1,4
+1.2
+0.4
+1
+0.3
+0.8
+0.2
+0.6
+0.1
+0.4Figure 4. Local distribution of |E| over the unit cell boundaries in a monolayer of golden spheres
+at λ = 5000 nm. (a) θ = 0; (b) θ = 45◦.
+Figure 5.
+Transmission and reflection spectra of a monolayer of golden spheres at normal and
+oblique incidence, θ = 0 and θ = 45◦.
+three distinct regions at both normal and oblique incidence. With decrease in λ from its
+maximum value 5000 nm to approximately 3200 nm, the optical properties of the monolayer
+change monotonically. Further, up to λ ≈ 1400 nm, there is the region of oscillations, where
+one can observe peaks and dips of R and T at wavelenghts that are nearly multiples of
+d = 500 nm. Those peaks and dips can be ascribed to the grating resonances occurred in
+8
+
+0.9
+transmittance
+0.8
+0.7
+*0=0T
+0.6
+-
+0-0=0°R
+and
+0.5
+含0=0°R+T
+Reflectance
+0.4
+0=45°, T
+0.3
++-0=45°R
+中0=45°R+T
+0.2
+0.1
+500
+1000
+1500
+2000
+2500
+入, nm(a)
+X107
+(q)
+X107
+7.4
+7
+7.2
+6.5
+7
+-
+6.8
+6
+6.6
+H
+5.5
+6.4
+6.2
+5Figure 6. Transmission and reflection spectra of a monolayer of golden inclusions “SRRs plus rods”
+at normal and oblique incidence, θ = 0 and θ = 45◦.
+periodic systems as a result of interaction between their structural elements excited by the
+incident wave. With further decrease in λ, the oscillations of R and T become more chaotic.
+As in the case of spherical inclusions, the layer of SRRs and rods looks translucent even at
+large enough wavelenghts: T + R < 0.6 at λ = 10d = 5000 nm.
+The peculiarities in the optical behavior of monolayers of golden spheres and SRRs with
+rods can also be observed in a thicker triple layer of golden spheres, see Fig. 7. In this case,
+notice the two peaks in R (at normal and oblique incidence) and the dip in T (at normal
+incidence) which are all located exactly at λ = 1000 nm, which is two times the lattice
+constant d. Obviously, they can be ascribed to the above mentioned grating resonances.
+IV.
+CONCLUSIONS
+The obtained results confirm the opinion [4] that any metamaterial homogenization
+method should be used with care in the intermediate operating regimes, when the metama-
+terial unit cell size is of the order of the operating wavelength. The value of the relative unit
+cell size d/λ at which homogenization methods fail to predict the optical properties of pe-
+riodic metamaterials depends on the geometry and material parameters of their inclusions.
+For the metamaterials we considered here, abrupt and substanial changes in the optical
+properties (as compared to their longwavelength values) occur at different values of d/λ:
+near 0.56 for the monolayer of golden spheres at oblique incidence, 0.4 for the triple layer of
+9
+
+0.9
+Reflectance and transmittance
+0.8
+0.7
+0.6
+*- 0=0°T
+0=0°,R
+0.5
+0=0°, R+T
+0.4
+0=45°T
+0.3
++
+0=45°.R
+0.2
+中
+0=45°R+T
+0.1
+0
+1000
+1500
+2000
+2500
+3000
+3500
+4000
+4500
+5000
+入,nmFigure 7. Transmission and reflection spectra of a triple layer of golden spheres at normal and
+oblique incidence, θ = 0 and θ = 45◦.
+golden spheres at normal incidence, and near 0.16 for the monolayer of SRRs with rods. For
+larger d/λ values, a crucial role in the optical properties formation play the diffraction and
+interference effects in the metamaterials, so the properties exhibit an oscillating behavior
+which cannot be predicted within the homogenization concept.
+[1] Pendry J.B., Holden A.J., Robbins D.J. and Stewart W.J. IEEE Trans. Microw. Theory Tech.
+47 2075–84 (1999).
+[2] D. Smith and J. Pendry, J. Opt. Soc. Am. B 23 391-403 (2006).
+[3] I. Tsukerman, J. Opt. Soc. Am. B 28 577–86 (2011).
+[4] V.V. Gozhenko, A.K. Amert, and K.W. Whites , New J. Phys. 15 043030 (2013).
+[5] S. Yoo et al., Nanophotonics 8 (6) 1063–1069 (2019).
+[6] O. Rybin and V. Khardikov, Optik 268 169768 (2022).
+10
+
+0.9
+Reflectance and transmittance
+0.8
+0.7
+0.6
+*
+0=0°T
+0=0°R
+0.5
+0=0°,R+T
+0.4
+0=45°T
+0.3
+0=45°R
+中
+0=45°R+T
+0.2
+0.1
+500
+1000
+1500
+入, nm

39E1T4oBgHgl3EQf6AV3/content/tmp_files/load_file.txt

@@ -0,0 +1,287 @@
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+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='03518v1  [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='optics]  9 Jan 2023  Size Effects in Periodic Metamaterials  Victor V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Gozhenko  Institute of Physics, Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' of Sciences of Ukraine,  46 Nauky Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', Kyiv 03680, Ukraine∗  1  Abstract  The optical properties of periodic electromagnetic metamaterials are considered as functions of  their relative unit cell size d/λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The reflection R and transmission T coefficients are numerically  calculated for some realistic metamaterials in a wide range of their relative unit cell size values  that comprises different operating regimes of the metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Peculiarities in R and T behavior  are discussed and the causes of those peculiarities are outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The obtained results support the  opinion on inapplicability of the very homogenization concept to metamaterials whose unit cell size  is comparable to the incident wavelength, in contrast to some previously published results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  INTRODUCTION  Most of the electromagnetic metamaterials are periodic structures, and their unit cells  consist of artificial inclusions designed to get a specific electromagnetic response (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', neg-  ative refraction or selective reflectivity) of the metamaterial sample as a whole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Periodic  metamaterial can be treated as a continuous and homogeneous medium if its unit cell size  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The concept of metamaterials homogenization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' A periodic metamaterial with unit cells  of size d is lit by an incident electromagnetic wave whose wavelength is λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' If d ≪ λ, then the  incident wave cannot “feel” the metamaterial inhomogenities, and metamaterial behave itself like  a continuous and homogeneous medium, whose parameters are εeff, µeff, neff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Representation of a  metamaterial by the corresponding homogeneous medium is correct if the optical properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=',  reflectance and transmittance) of the metamaterial and the medium are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  ∗ vigo@iop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='kiev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='ua  2  reflected  transmitted  Eeff  Ueff  neff  incidentd (the lattice constant) is much smaller then its operating wavelength λ, d ≪ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' In such a  case, the metamaterial can be characterized by its effective parameters—the effective per-  mittivity εeff, permeability µeff, and index of refraction (the refractive index) neff = √εµ, see  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Calculation of the effective parameters values for a given metamaterial (with given shape,  size, and material of its inclusions, as well as the size and geometry of its unit cell) is im-  portant, for example, for predicting the optical properties the metamaterial will reveal in  experiments and applications, and is based on calculating the local electric and magnetic  fields within the unit cell and proper averaging of those fields over the cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Sometimes—in  case of simple inclusions—it can be done analytically;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' in general case, numerical computa-  tions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  To facilitate calculation of the effective parameters of periodic metamaterials, a number  of homogenization theories and methods were proposed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 1–6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Those methods  differ from each other by, particularly, the way they calculate the average values of the local  fields in the metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  In metamaterials applications, condition d ≪ λ (say, d = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='01λ) is not always met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' For  example, optical applications (where λ ≈ 400 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 800 nm) implies that the unit cell size  should be of the order of 50 nm or less.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' However, metamaterials with d = 200 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 300 nm  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', d/λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='25 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5) are often used there because the less the unit cell size, the more  expensive manufacturing process of the metamaterial sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' On the other hand, some  applications—most notably the negative index of refraction—require for a metamaterial to  work in the resonant regime (where the negative n is achieveable) meaning d/λ ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Therefore, some authors tried to elaborate homogenization methods suitable for an extended  range of d/λ values, and not only for small d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Some of them believe that their methods are  valid for metamaterials with substantial or even arbitrary unit cell size (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 1, 3, and  6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Strictly speaking, any homogenization method can give plausible results for metamate-  rials working in the long wavelength (quasistatic) regime only, where the condition d ≪ λ  is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' In the opposite case of short (relative to the unit cell size) waves, d ≫ λ, the  very homogenization concept should fail, and metamaterials cannot be treated as homoge-  neous media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' In this case, propagation of incident waves through a metamaterial obeys the  geometrical optics laws, and reflection of an incident wave from the metamaterial inclusions  3  plays a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Last, in the intermediate regime, where d ≈ λ, homogenization meth-  ods should not work since they do not account for the diffraction effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', the Bragg’s  reflection) which are significant in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Earlier [4], it was shown that different homogenization methods give more and more di-  verging results as the relative unit cell size of metamaterials increases from zero to approx-  imately d/λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' At larger d/λ values, calculations of homogenized effective parameters  of a metamaterial can still formally be performed, but those parameters cannot describe  correctly the optical properties of the metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  In the present paper, the optical properties of periodic metamaterials are considered in a  wider range of their relative unit cell sizes d/λ, and the effects of the cell size on the optical  behaviour of the matematerials is discussed in more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  BASIC FORMULAE  We are interested in calculating the observable quantities—transmittance T and re-  flectance R of a metamaterial—which are dimensionless coefficients defined as  T = Itr  I0  ,  R = Iref  I0  ,  where I0, Itr, Iref are the intensities of incident, transmitted through the metamaterial, and  reflected waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The intensities are the energy flux densities of the corresponding waves and  can be calculated as time-averaged values of the Poynting vector S of those waves (indices  are omitted below for simplicity):  I = ⟨S⟩ = 1  τ  t+τ  �  t  S(t′)dt′  In case of monochromatic incident plane wave  E = E0e[i(k·r−ωt)],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  (1)  H = H0e[i(k·r−ωt)]  (2)  all the waves involved are also monochromatic and their intensities can be calculated from  I = 1  2Re(E × H∗),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  4  where the asterisk denotes the complex conjugation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  From the energy conservation law  applied to the interaction of electromagnetic waves with a lossy medium, it follows  T + R + A = 1,  where A is the absorptance (the absorption coefficient) of the medium which define the rate  of electromagnetic energy absorption inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' If T and R values for a medium are known  (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', are experimentally measured or theoretically calculated), its absorptance can be found  as  A = 1 − T − R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  NUMERICAL RESULTS AND DISCUSSION  Numerical simulations are carried out for metamaterials with cubic lattice and consisted of  inclusions of various shapes that are often used in metamaterial science and applications—  spheres, rods, Split-Ring Resonators (SRRs), and Ω-like inclusions (“omegas”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Optical  reflectance R and transmittance T of the simulated metamaterials are calculated in a wide  range of their relative unit cell size d/λ at several values of the incidence angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Calculations  of R and transmittance T are complemented with the local E field distributions across  the unit cells in different regimes the metamaterials operate in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' All the calculations are  performed with COMSOL Multiphysics software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Presented below are exemplary calculation  results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Schematics of the unit cells of the simulated metamaterials are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The  materials are primarily infinite monolayers of thickness d and that in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 2d is a triple layer  of thickness 3d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The unit cell size d = 500 nm is the same for all the materials and remains  unchangeable in all the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Variations in the relative unit cell size d/λ are made  by changing the wavelength λ of the incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 3 one can see that all the inclusions give a prominent response to the incident  wave even at normal incidence, the response of spherical inclusions being dipole-like (see  panel (a), the field distribution in the middle plane), while those of “SRRs with rods” system  and “omegas” are more tricky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  The incidence angle θ also affects the electromagnetic field distribution inside metama-  terials: even for simple spherical inclusions in the quasistatic regime (λ = 10d) the electric  5  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Unit cells of the simulated metamaterials: (a) a monolayer of spherical particles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' (b) a  monolayer of omegas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' (c) a monolayer of SRRs and rods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' (d) a triple layer of spherical particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  All the inclusions are made of gold, and the size of the individual unit cells is d = 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  fields in the unit cell differ substantially at normal and oblique incidence, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 5–7 are the transmission and reflection spectra numerically calculated for  the metamaterials depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 2a,c,d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  According to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 5, at normal incidence of a long enough wave with λ = 3000 nm (which  is six times the unit cell size d) onto the monolayer of golden spheres, the metamaterial  behaves itself as a transparent sheet: its reflection coefficient R in this regime is close to  zero, and its transmission coefficent T is near unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' If λ decreases and approaches the unit  cell size d, the metamaterial gradually looses its transparency and becomes more and more  reflective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The reflection coefficient has its maximum R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='45 at λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2d = 600 nm,  and the transmission coefficient is minimal (T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='3) at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Further decrease of λ  results in monotonic increase of the material transparency (with the peak value T ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='77 at  λ = d = 500 nm) and monotonic decrease of its reflectance up to R ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The radical change  in the behavior of R and T at λ ≤ 600 nm (when d/λ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='83) is probably due to the onset  of the diffraction and interference effects in the periodic metamaterial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Note that the sum T + R becomes distinctly less then unity at λ ≤ 650 nm, or d/λ ≥  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' It means that light absorption by the metamaterial is substantial in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The  minimum value of the absorption coefficient A = 1 − T + R ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='67 is observed at λ = 550  nm, which implies that about one third of the incident energy flux is absorbed by the  metamaterial inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  The absorption can be ascribed to electric currents induced in  individual inclusions (golden spheres) by the incident wave as well scattered waves from  neighboring particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Analogous behavior is observed at oblique incidence (see the respective curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 5  6  (a)  (b)  (c)  (d)Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Local distributions of the absolute value of the electric field E inside the unit cells of the  metamaterials shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 2a–c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Left column, the field on the unit cell boundaries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' right column,  the field in the middle plane of the cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' λ = 1000 nm, normal incidence (θ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Directions of  E and H vectors of the incident wave are depicted in panel (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Distribution of |E| allows one to  easily determine those places where the electric energy is concentrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  for the case of θ = 45◦): with decreasing λ, the region of insufficient changes in R, T passes  (starting from λ ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8d = 900 nm) to the region of their abrupt changes and oscillating  behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Note, however, that the metamaterial in this case is semi-transparent even at  large λ: R + T ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='62 at λ = 3000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  The plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 6 refer to the monolayer of golden inclusions “SRR plus rod” and have  7  (a)  X107  X108  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='3  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='1  6  1  E  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='9  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  H  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='7  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  (q)  X108  X108  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  (c)  X108  X108  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='9  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='7  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  1,4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='1  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Local distribution of |E| over the unit cell boundaries in a monolayer of golden spheres  at λ = 5000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' (a) θ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' (b) θ = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  Transmission and reflection spectra of a monolayer of golden spheres at normal and  oblique incidence, θ = 0 and θ = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  three distinct regions at both normal and oblique incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' With decrease in λ from its  maximum value 5000 nm to approximately 3200 nm, the optical properties of the monolayer  change monotonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Further, up to λ ≈ 1400 nm, there is the region of oscillations, where  one can observe peaks and dips of R and T at wavelenghts that are nearly multiples of  d = 500 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Those peaks and dips can be ascribed to the grating resonances occurred in  8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='9  transmittance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='7  0=0T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6 0-0=0°R  and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  含0=0°R+T  Reflectance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  0=45°, T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='3  +-0=45°R  中0=45°R+T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='1  500  1000  1500  2000  2500  入, nm(a)  X107  (q)  X107  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  7  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  6  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  H  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  5Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Transmission and reflection spectra of a monolayer of golden inclusions “SRRs plus rods”  at normal and oblique incidence, θ = 0 and θ = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  periodic systems as a result of interaction between their structural elements excited by the  incident wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' With further decrease in λ, the oscillations of R and T become more chaotic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  As in the case of spherical inclusions, the layer of SRRs and rods looks translucent even at  large enough wavelenghts: T + R < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6 at λ = 10d = 5000 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  The peculiarities in the optical behavior of monolayers of golden spheres and SRRs with  rods can also be observed in a thicker triple layer of golden spheres, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' In this case,  notice the two peaks in R (at normal and oblique incidence) and the dip in T (at normal  incidence) which are all located exactly at λ = 1000 nm, which is two times the lattice  constant d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Obviously, they can be ascribed to the above mentioned grating resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  CONCLUSIONS  The obtained results confirm the opinion [4] that any metamaterial homogenization  method should be used with care in the intermediate operating regimes, when the metama-  terial unit cell size is of the order of the operating wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' The value of the relative unit  cell size d/λ at which homogenization methods fail to predict the optical properties of pe-  riodic metamaterials depends on the geometry and material parameters of their inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  For the metamaterials we considered here, abrupt and substanial changes in the optical  properties (as compared to their longwavelength values) occur at different values of d/λ:  near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='56 for the monolayer of golden spheres at oblique incidence, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4 for the triple layer of  9  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='9  Reflectance and transmittance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6  *- 0=0°T  0=0°,R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  0=0°, R+T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  0=45°T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='3  +  0=45°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  中  0=45°R+T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='1  0  1000  1500  2000  2500  3000  3500  4000  4500  5000  入,nmFigure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' Transmission and reflection spectra of a triple layer of golden spheres at normal and  oblique incidence, θ = 0 and θ = 45◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  golden spheres at normal incidence, and near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='16 for the monolayer of SRRs with rods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=' For  larger d/λ values, a crucial role in the optical properties formation play the diffraction and  interference effects in the metamaterials, so the properties exhibit an oscillating behavior  which cannot be predicted within the homogenization concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='  [1] Pendry J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content=', Holden A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
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+page_content='  10  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='9  Reflectance and transmittance  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='7  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='6 0=0°T  0=0°R  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='5  0=0°,R+T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='4  0=45°T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='3  0=45°R  中  0=45°R+T  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}
+page_content='1  500  1000  1500  入, nm' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/39E1T4oBgHgl3EQf6AV3/content/2301.03518v1.pdf'}

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+arXiv:2301.00745v1  [math.DG]  2 Jan 2023
+Moduli of triples of points in quaternionic hyperbolic geometry
+Igor Almeida
+Nikolay Gusevskii∗
+[email protected]
+[email protected]
+Departamento de Matem´atica
+Universidade Federal de Minhas Gerais
+Belo Horizonte – MG
+Brazil
+30123-970
+Abstract
+In this work, we describe the moduli of triples of points in quaternionic projective space which de-
+fine uniquely the congruence classes of such triples relative to the action of the isometry group of
+quaternionic hypebolic space Hn
+Q. To solve this problem, we introduce some basic invariants of triples
+of points in quaternionic hyperbolic geometry.
+In particular, we define quaternionic analogues of
+the Goldman invariants for mixed configurations of points introduced by him in complex hyperbolic
+geometry.
+MSC: 32H20; 20H10; 22E40; 57S30; 32G07; 32C16
+Keywords: Quaternionic hyperbolic space. Moduli of triples.
+Introduction
+The purpose of this paper is to describe some numerical invariants associated to an ordered triple of points
+in quaternionic projective space. These invariants describe the equivalence classes of such triples relative
+to the action of the isometry group of quaternionic hypebolic space Hn
+Q. We give a construction of the
+quaternionic angular invariant, an analogue of the Cartan invariant in complex hyperbolic geometry, see
+[8], which parametrizes triples of isotropic points. Also, we represent a quaternionic analogue of Brehm’s
+shape invariant, see [4], in complex hyperbolic geometry, which is used to parametrize triples of points
+in Hn
+Q. Then we define a quaternionic analogue of the Goldman η-invariant for mixed configurations of
+points introduced by him in complex hyperbolic geometry to study the intersection of bisectors, see [13].
+Using these invariants, we describe the moduli of the corresponding triples relative to the action of the
+isometry group of quaternionic hypebolic space Hn
+Q. In order to solve the congruence problems, we use the
+methods related to Gram matrices of configurations of points developed in complex hyperbolic geometry
+in [4], [5], [10], [11], [12], [15], [16]. In this work, we describe the moduli of all possible configurations
+of three points in quaternionic projective space of any dimension and give a geometric interpretation of
+them.
+We remark that some of these problems were considered by Cao, see [7]. Unfortunately, some of the
+main results of this work are not correct as stated, see Theorem 1.1 (items (ii) and (iii)) in [7]. We provide
+the corresponding counter-examples, see Section 2.4.1 and Section 2.4.2.
+∗Corresponding author.
+1
+
+The work is organized as follows. In Section 1, we summarize some basic results about geometry of
+quaternionic hyperbolic space. In Section 2, we describe the moduli of triples of points in quaternionic
+hyperbolic geometry.
+1
+Preliminaries
+In this section, we recall some basic results related to quarternions and geometry of projective and hyper-
+bolic spaces.
+1.1
+Quaternions
+First, we recall some basic facts about the quaternions we need. The quaternions Q are the R-algebra
+generated by the symbols i, j, k with the relations
+l2 = j2 = k2 = −1,
+ij = −ji = k,
+jk = −kj = 1,
+ki = −ik = j.
+So, Q is a skew field and a 4-dimensional division algebra over the reals.
+Let a ∈ Q. We write a = a0 + a1i + a2j + a3k, ai ∈ R, then by definition
+¯a = a0 − a1i − a2j − a3k,
+Re a = a0,
+Im a = a1i + a2j + a3k.
+Note that, in contrast with the complex numbers, Im a is not a real number (if ai ̸= 0 for some
+i = 1, 2, 3), and that conjugation obeys the rule
+ab = ¯b¯a.
+Also, we define |a| = √a¯a. We have that if a ̸= 0 then a−1 = ¯a/|a|2.
+In what follows, we will identify the reals numbers R with R1 and the complex numbers C with the
+subfield of Q generated over R by 1 and i.
+Two quaternions a and b are called similar if there exists λ ̸= 0 such that a = λbλ−1. By replacing λ
+by τ = λ/|λ|, we may always assume λ to be unitary.
+The following proposition was proved in [6].
+Proposition 1.1 Two quaternions a and b are similar if and only if Re a = Re b and |a| = |b|. Moreover,
+every similarity class contains a complex number, unique up to conjugation.
+Corollary 1.1 Any quaternion a is similar to a unique complex number b = b0 + b1i such that b1 ≥ 0.
+Also, this proposition implies that every quaternion is similar to its conjugate.
+Example: jzj−1 = ¯z for all z ∈ C.
+We say that a ∈ Q is imaginary if Re(a) = 0. Let us suppose that a is imaginary and |a| = 1. Then
+a2 = −1. This implies that the real span of 1 and a is a subfield of Q isomorphic to the field of complex
+numbers. We denote this subfield by C(a). It is easy to prove that any subfield of Q containing real
+numbers and isomorphic to the field of complex numbers is of the form C(a) for some a imaginary with
+|a| = 1.
+The following was proved in [9].
+2
+
+Proposition 1.2 Let a be as above. Then the centralizer of a is C(a).
+More generally, for any λ ∈ Q, let C(λ) also denote the real span of 1 and λ.
+Proposition 1.3 Let λ ∈ Q \ R. Then the centralizer of λ is C(λ).
+1.2
+Hyperbolic spaces
+In this section we discuss two models for the hyperbolic spaces, its isometry group and totally geodesics
+submanifolds.
+1.2.1
+Projective Model
+We denote by F one of the real division algebras R, C, or Q. Let us write Fn+1 for a right F - vector space
+of dimension n + 1. The F- projective space PFn is the manifold of right F-lines in Fn+1. Let π denote a
+natural projection from Fn+1 \ {0} to the projective space PFn.
+Let Fn,1 denote a (n + 1)-dimensional F-vector space equipped with a Hermitian form Ψ = ⟨−, −⟩ of
+signature (n, 1). Then there exists a (right) basis in Fn+1 such that the Hermitian product is given by
+⟨v, w⟩ = v∗Jn+1w, where v∗ is the Hermitian transpose of v and Jn+1 = (aij) is the (n+1)×(n+1)-matrix
+with aij = 0 for all i ̸= j, aii = 1 for all i = 1, . . . , n, and aii = −1 when i = n + 1.
+That is,
+⟨v, w⟩ = ¯v1w1 + . . . + ¯vnwn − ¯vn+1wn+1,
+where vi and wi are coordinates of v and w in this basis. We call such a basis in Fn,1 an orthogonal basis
+defined by a Hermitian form Ψ = ⟨−, −⟩.
+Let V−, V0, V+ be the subsets of Fn,1\{0} consisting of vectors where ⟨v, v⟩ is negative, zero, or positive
+respectively. Vectors in V0 are called null or isotropic, vectors in V− are called negative, an vectors in V+
+are called positive. Their projections to PFn are called isotropic, negative, and positive points respectively.
+The projective model of hyperbolic space Hn
+F is the set of negative points in PFn, that is, Hn
+F = π(V−).
+We will consider Hn
+F equipped with the Bergman metric [9]:
+d(p, q) = cosh−1{|Ψ(v, w)|[Ψ(v, v)Ψ(w, w)]−1/2},
+where p, q ∈ Hn
+F, and π(v) = p, π(w) = q.
+The boundary ∂Hn
+F = π(V0) of Hn
+F is the sphere formed by all isotropic points.
+Let U(n, 1; F) be the unitary group corresponding to this Hermitian form Φ. If g ∈ U(n, 1; F), then
+g(V−) = V− and g(vλ) = (g(v))λ, for all λ ∈ F. Therefore U(n, 1; F) acts in PFn, leaving Hn
+F invariant.
+The group U(n, 1; F) does not act effectively in Hn
+F. The kernel of this action is the center Z(n, 1; F).
+Thus, the projective group PU(n, 1; F) = U(n, 1; F)/Z(n, 1; F) acts effectively. The center Z(n, 1, F) in
+U(n, 1; F) is {±E} if F = R or Q, and is the circle group {λE : |λ| = 1} if F = C. Here E is the identity
+transformation of Fn,1.
+It is well-known, see for instance [9], that PU(n, 1; F) acts transitively in Hn
+F and doubly transitively
+on ∂Hn
+F.
+We remark that
+• if F = R then Hn
+F is a real hyperbolic space Hn
+R,
+3
+
+• if F = C then Hn
+F is a complex hyperbolic space Hn
+C,
+• if F = Q then Hn
+F is a quaternionic hyperbolic space Hn
+Q.
+It is easy to show [9] that H1
+Q is isometric to H4
+R.
+1.2.2
+The ball model
+In this section, we consider the space Fn,1 equipped by an orthogonal basis
+e = {e1, . . . , en, en+1}.
+For any v ∈ Fn,1, we write v = (z1, . . . , zn, zn+1), where zi, i = 1, . . . , n + 1 are coordinates of v in this
+basis.
+If v = (z1, . . . , zn, zn+1) ∈ V−, the condition ⟨v, v⟩ < 0 implies that zn+1 ̸= 0. Therefore, we may define
+a set of coordinates w = (w1, . . . , wn) in Hn
+F by wi(π(z)) = ziz−1
+n+1. In this way Hn
+F becomes identified with
+the ball
+B = B(F) = {w = (w1, . . . , wn) ∈ Fn : Σn
+i=1|wi|2 < 1}.
+With this identification the map π : V− → Hn
+F has the coordinate representation π(z) = w, where
+wi = ziz−1
+n+1.
+1.2.3
+Totally geodesic submanifolds
+We will need the following result, see [9], which describes all totally geodesic submanifolds of Hn
+F.
+Let F be a subfield of F.
+An F-unitary subspace of Fn,1 is an F-subspace of Fn+1 in which the
+Hermitian form Φ is F-valued. An F-hyperbolic subspace of Fn,1 is an F-unitary subspace in which the
+Hermitian form Φ is non-degenerate and indefinite.
+Proposition 1.4 Let M be a totally geodesic submanifold of Hn
+F. Then either
+(a) M is the intersection of the projectivization of an F-hyperbolic subspace of Fn,1 with Hn
+F for some
+subfield F of F, or
+(b) F = Q, and M is a 3-dimensional totally geodesic submanifold of a totally geodesic quaternionic
+line H1
+Q in Hn
+Q.
+From the last proposition follows that
+• in the real hyperbolic space Hn
+R any totally geodesic submanifold is isometric to Hk
+R, k = 1, . . . , n,
+• in the complex hyperbolic space Hn
+C any totally geodesic submanifod is isometric to Hk
+C, or to Hk
+R,
+k = 1, . . . , n,
+• in the quaternionic hyperbolic space Hn
+Q any totally geodesic submanifold is isometric to Hk
+Q, or to
+Hk
+C, or to Hk
+R, k = 1, . . . , n, or to a 3-dimensional totally geodesic submanifold of a totally geodesic
+quaternionic line H1
+Q.
+In what follows we will use the following terminology:
+4
+
+• A totally geodesic submanifold of Hn
+Q isometric to H1
+Q is called a quaternionic geodesic.
+• A totally geodesic submanifold of Hn
+Q isometric to H1
+C is called a complex geodesic.
+• A totally geodesic submanifold of Hn
+Q isometric to H2
+R is called a real plane.
+It is clear that two distinct points in Hn
+Q ∪ ∂Hn
+F span a unique quaternionic geodesic. We also remark
+that any 2-dimensional totally geodesic submanifold of a totally geodesic quaternionic line H1
+Q is isometric
+to H1
+C.
+Proposition 1.5 Let V be a subspace of Fn,1. Then each linear isometry of V into Fn,1 can be extended
+to an element of U(n, 1; F).
+This is a particular case of the Witt theorem, see [17].
+Corollary 1.2 Let S ⊂ Hn
+F be a totally geodesic submanifold. Then each linear isometry of S into Hn
+F
+can be extended to an element of the isometry group of Hn
+F.
+An interesting class of totally geodesic submanifolds of the quaternionic hyperbolic space Hn
+Q are
+submanifods which we call totally geodesic submanifolds of complex type, or simply, submanifolds of
+complex type. Their construction is the following. Let Cn+1(a) ⊂ Qn+1 be the subset of vectors in Qn+1
+with coordinates in C(a), where a is a imaginary quaternion, |a| = 1. Then Cn+1(a) is a vector space
+over the field C(a). The projectivization of Cn+1(a), denoted by Mn(C(a)), is a projective submanifold
+of PQn of real dimension 2n. We call this submanifold Mn(C(a)) a projective submanifold of complex
+type of maximal dimension. It is clear that the space Cn+1(a) is indefinite. The intersection Mn(C(a))
+with Hn
+Q is a totally geodesic submanifold of Hn
+Q, called a totally geodesic submanifold of complex type of
+maximal dimension. It was proven in [9] that all these submanifolds are isometric, and, moreover, they
+are globally equivalent with respect to the isometry group of Hn
+Q, that is, for any two such submanifolds M
+and N there exists an element g ∈ PU(n, 1; Q) such that M = g(N). In particular, all of them are globally
+equivalent with respect to PU(n, 1; Q) to the canonical totally geodesic complex submanifold Hn
+C defined
+by Cn+1 ⊂ Qn+1. This corresponds to the canonical subfield of complex numbers C = C(i) ⊂ Q in the
+above.
+If V k+1 ⊆ Cn+1(a) is a subspace of complex dimension k + 1, then its projectivization W is called a
+projective submanifold of complex type of complex dimension k. When V k+1 ⊆ Cn+1, then its projec-
+tivization W is called a canonical projective submanifold of complex type of complex dimension k In this
+case, we will denote W as PCk.
+If V k+1 ⊆ Cn+1(a) is indefinite, then the intersection of its projectivization with Hn
+Q is a totally
+geodesic submanifold of Hn
+Q. We call this submanifold of Hn
+Q a totally geodesic submanifold of complex
+type of complex dimension k. When V k+1 ⊆ Cn+1, we call this totally geodesic submanifold a canonical
+totally geodesic submanifold of complex type of complex dimension k, or a canonical complex hyperbolic
+submanifold of dimension k of Hn
+Q. In this case, we will denote this submanifold as Hk
+C.
+1.2.4
+A little more about the isometry group of the quaternionic hyperbolic space
+Let us consider the complex hyperbolic space Hn
+C.
+It has a natural complex structure related to its
+isometry group, and the isometry group of Hn
+Q is generated by the holomorphic isometry group, which is
+the projective group PU(n, 1; C), and the anti-holomorphic isometry σ induced by complex conjugation
+in Cn+1. This anti-holomorphic isometry corresponds to the unique non-trivial automorphism of the field
+of complex numbers. Below we consider a similar isometry of quaternionic hyperbolic space Hn
+Q.
+5
+
+We recall that if f : Q → Q is an automorphism of Q, then f is an inner automorphism of Q, that is,
+f(q) = aqa−1 for some a ∈ Q, a ̸= 0.
+It follows from the fundamental theorem of projective geometry, see [2], that each projective map
+L : PQn → PQn is induced by a semilinear or linear map ˜L : Qn+1 → Qn+1.
+It is easy to see that if a projective map
+L : PQn → PQn
+is induced by a semilinear map
+˜L : Qn+1 → Qn+1, ˜L(v) = ava−1, v ∈ Qn+1, a ∈ Q \ R,
+then it is also induced by a linear map v �→ av.
+Therefore, the projective group of PQn is the projectivization of the linear group of Qn+1.
+This implies that if
+L : Hn
+Q → Hn
+Q
+is an isometry, then L is induced by a linear isometry
+˜L : Qn,1 → Qn,1.
+This explains why the group of all isometries of Hn
+Q is the projectivization of the linear group U(n, 1; Q),
+that is, PU(n, 1; Q).
+Next we consider a curious map, which is an isometry of the quaternionic hyperbolic space, that has
+no analogue in geometries over commutative fields. Let ˜La : v �→ av, v ∈ Qn+1, a ∈ Q, a is not real.
+The projectivization of this linear map defines a non-trivial map La : PQn → PQn. We remark that in
+projective spaces over commutative fields this map La is identity. It easy to see that ˜La ∈ U(n, 1; Q) if
+and only if |a| = 1, so La in PU(n, 1; Q) if and only if |a| = 1.
+Proposition 1.6 Let a ∈ Q\R, |a| = 1. Then the fixed point set Sa of La is a totally geodesic submanifold
+of complex type of maximal dimension in Hn
+Q. This submanifold is globally equivalent to the canonical
+complex hyperbolic submanifold Hn
+C of Hn
+Q.
+Proof: The proof follows from Proposition 1.3.
+It is easy to see that if a is imaginary, then La is an involution. We call this isometry La a geodesic
+reflection in Sa.
+2
+Moduli of triples of points in quaternionic projective space
+In this section, we describe numerical invariants associated to an ordered triple of points in PQn which
+define the equivalence class of a triple relative to the diagonal action of PU(n, 1; Q).
+6
+
+2.1
+The Gram matrix
+Let p = (p1, . . . , pm) be an ordered m-tuple of distinct points in PQn of quaternionic dimension n ≥ 1.
+Then we consider a Hermitian quaternionic m × m-matrix
+G = G(p, v) = (gij) = (⟨vi, vj⟩),
+where v = (v1, . . . , vm), vi ∈ Qn,1, π(vi) = pi, is a lift of p.
+We call G a Gram matrix associated to a m-tuple p defined by v. Of course, G depends on the chosen
+lifts vi. When replacing vi by viλi, λi ∈ Q, λi ̸= 0, we get ˜G = D∗ GD, where D is a diagonal quaternionic
+matrix, D = diag(λ1, . . . , λm),
+We say that two Hermitian quaternionic m × m - matrices H and ˜H are equivalent if there exists a
+diagonal quaternionic matrix D = diag(λ1, . . . , λm), λi ̸= 0, such that ˜H = D∗ H D.
+Thus, to each ordered m-tuple p of distinct points in PQn is associated an equivalence class of Hermitian
+quaternionic m × m - matrices.
+Proposition 2.1 Let p = (p1, · · · , pm) be an ordered m-tuple of distinct negative points in PQn. Then
+the equivalence class of Gram matrices associated to p contains a matrix G = (gij) such that gii = −1 and
+g1j = r1j are real positive numbers for j = 2, . . . , m.
+Proof: Let v = (v1, . . . , vm) be a lift of p. Since the vectors vi are negative, we have that gij ̸= 0
+for all i, j = 1, . . . , m, see, for instance, [17].
+First, by appropriate re-scaling, we may assume that
+gii = ⟨vi, vi⟩ = −1. Indeed, since ⟨vi, vi⟩ < 0, then λi = 1/
+�
+−⟨vi, vi⟩ is well defined. Since λi ∈ R, we
+have that
+⟨viλi, viλi⟩ = λ2
+i ⟨vi, vi⟩ = ⟨vi, vi⟩/|⟨vi, vi⟩| = −1.
+Then we get the result we need by replacing the vectors vi, i = 2, . . . , m, if necessarily, by viλi, where
+λi = ⟨v1, vi⟩/|⟨v1, vi⟩|.
+Indeed, since |λi| = 1, we have that ⟨viλi, viλi⟩ = −1, i = 2, . . . , m. On the other hand, for all i > 1
+⟨v1, viλi⟩ = ⟨v1, vi⟩λi = |⟨v1, vi⟩| > 0.
+Let p and q be two points in PQn. We say that p and q are orthogonal if ⟨v, w⟩ = 0 for some lifts v
+and w of p and q respectively. It is clear that if p and q are orthogonal then ⟨v, w⟩ = 0 for all lifts v and
+w of p and q.
+Let p = (p1, . . . , pm) be an ordered m-tuple of distinct points in PQn. We call p generic if pi and pj
+are not orthogonal for all i, j = 1, . . . , m, i ̸= j.
+Let G = (gij) be a Gram matrix associated to p. Then p is generic if and only if gij ̸= 0 for all
+i, j = 1, . . . , m, i ̸= j.
+Proposition 2.2 Let p = (p1, · · · , pm) be an ordered generic m-tuple of distinct positive points in PQn.
+Then the equivalence class of Gram matrices associated to p contains a matrix G = (gij) such that gii = 1
+and g1j = r1j are real positive numbers for j = 2, . . . , m.
+7
+
+Proof: The proof is a slight modification of the proof of Proposition 2.1.
+It is easy to see that a matrix G = (gij) defined in Propositions 2.1 and 2.2 is unique. We call this
+matrix G a normal form of the associated Gram matrix. Also, we call G the normalized Gram matrix.
+We recall that a subspace V ⊂ Fn,1 is called singular or degenerate if it contains at least one non-zero
+vector that is orthogonal to all vectors in V . Otherwise, V is called regular.
+Remark 2.1 It is easy to see that if V is singular then V contains at least one isotropic vector and does
+not contain negative vectors.
+Lemma 2.1 Let V = {v1, . . . , vm} and W = {w1, . . . , wm} be two subspaces of Qn,1 spanned by vi and wi.
+Suppose that V and W are regular, and ⟨vi, vj⟩ = ⟨wi, wj⟩, for all i, j = 1, . . . m. Then the correspondence
+vi �→ wi can be extended to an isometry of Qn,1.
+Proof: The proof follows from Theorem 1 in [14].
+Proposition 2.3 Let p = (p1, . . . , pm) and p′ = (p′
+1, . . . , p′
+m) be two ordered m-tuples of distinct negative
+points in PQn. Then p and p′ are congruent relative to the diagonal action of PU(n, 1; Q) if and only if
+their associated Gram matrices are equivalent.
+Proof: Let V and V ′ be the subspaces spanned by vi and v′
+i, i = 1, . . . , m. Then it is clear that V and
+V ′ are regular. Since all the points pi are distinct, Lemma 2.1 implies that the map defined by v �→ v′
+extends to a linear isometry of Qn,1. The projectivization of this isometry maps p in p′.
+Corollary 2.1 Let p = (p1, . . . , pm) and p′ = (p′
+1, . . . , p′
+m) be two ordered m-tuples of distinct negative
+points in PQn. Then p and p′ are congruent relative to the diagonal action of PU(n, 1; Q) if and only if
+their normalized Gram matrices are equal.
+By applying the similar arguments, we get the following
+Proposition 2.4 Let p = (p1, . . . , pm) and p′ = (p′
+1, . . . , p′
+m) be two ordered generic m-tuples of distinct
+positive points in PQn such that the subspaces V and V ′ spanned by some lifts of p and p′ are regular.
+Then p and p′ are congruent relative to the diagonal action of PU(n, 1; Q) if only if their associated Gram
+matrices are equivalent.
+Corollary 2.2 Let p = (p1, . . . , pm) and p′ = (p′
+1, . . . , p′
+m) be two ordered generic m-tuples of distinct
+positive points in PQn such that the subspaces V and V ′ spanned by some lifts of p and p′ are regular.
+Then p and p′ are congruent relative to the diagonal action of PU(n, 1; Q) if and only if their normalized
+Gram matrices are equal.
+Remark 2.2 It is easy to see that a subspace V in Qn,1 is singular if and only if its projectivization is
+a projective submanifold of PQn tangent to ∂Hn
+Q at an unique isotropic point p, lying, except this point p,
+in the positive part of PQn.
+2.2
+Invariants of triangles in quaternionic hyperbolic geometry
+In this section, we define some invariants of ordered triples of points in PQn which generalize Cartan’s
+angular invariant and Brehm’s shape invariants in complex hyperbolic geometry to quaternionic hyperbolic
+geometry.
+8
+
+2.2.1
+Quaternionic Cartan’s angular invariant
+First, we recall the definition of Cartan’ s angular invariant in complex hyperbolic geometry.
+Let p = (p1, p2, p3) be an ordered triple of points in ∂Hn
+C. Then Cartan’s invariant A(p) of p is defined
+as
+A(p) = arg(−⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩),
+where vi is a lift of pi.
+It is easy to see that A(p) is well-defined, that is, it is independent of the chosen lifts, and it satisfies
+the inequality
+−π/2 ≤ A(p) ≤ π/2.
+The inequalities above follow from the following proposition, see [13].
+Proposition 2.5 Let v, w, u ∈ C2,1 be isotropic or negative vectors, then
+Re(⟨v, w⟩⟨w, u⟩⟨u, v⟩) ≤ 0.
+Remark 2.3 It is possible to extend the Cartan invariant of triples of isotropic points to triples of points
+in Hn
+C ∪ ∂Hn
+C. Indeed, no difficulty arises in the above definition because ⟨v, w⟩ ̸= 0 for any v, w ∈ V0 ∪ V−.
+Cartan’s invariant is the only invariant of an ordered triple of isotropic points in the following sense:
+Proposition 2.6 Let p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) be two ordered triples of distinct points in ∂Hn
+C.
+Then p and p′ are congruent relative to the diagonal action of PU(n, 1; C) if and only if A(p) = A(p′).
+The Cartan angular invariant A enjoys also the following properties, see [13]:
+1. If σ is a permutation, then
+A(pσ(1), pσ(2), pσ(3)) = sign(σ)A(p1, p2, p3),
+2. Let p = (p1, p2, p3) be an ordered triple of points in ∂Hn
+C. Then p lies in the boundary of a complex
+geodesic in Hn
+C if and only if A(p) = ±π/2,
+3. Let p = (p1, p2, p3) be an ordered triple of points in ∂Hn
+C. Then p lies in the boundary of a real plane
+in Hn
+C if and only if A(p) = 0,
+4. Cocycle property. Let p1, p2, p3, p4 be points in Hn
+C ∪ ∂Hn
+C. Then
+A(p1, p2, p3) + A(p1, p3, p4) = A(p1, p2, p4) + A(p2, p3, p4).
+5. If g ∈ PU(n, 1; C) is a holomorphic isometry, then A(g(p)) = A(p), and if g is an anti-holomorphic
+isometry, then A(g(p)) = −A(p).
+Next we define Cartan’s angular invariant in quaternionic hyperbolic geometry.
+Let v = (v1, v2, v3) be an ordered triple of vectors in Qn,1. Then
+H(v1, v2, v3) = ⟨v1, v2, v3⟩ = ⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩
+is called the Hermitian triple product. An easy computation gives the following.
+9
+
+Lemma 2.2 Let wi = viλi, λi ∈ Q, λi ̸= 0, then
+H(w1, w2, w3) = ⟨w1, w2, w3⟩ = λ1H(v1, v2, v3)λ1|λ2|2|λ3|2 =
+λ1
+|λ1|H(v1, v2, v3) λ1
+|λ1||λ1|2|λ2|2|λ3|2.
+Corollary 2.3 Let p = (p1, p2, p3) be an ordered triple of distinct points, pi ∈ PQn. Then there exists a
+lift v = (v1, v2, v3) of p = (p1, p2, p3) such that H(v1, v2, v3) is a complex number.
+Proof: The proof follows by applying Lemma 2.2, Proposition 1.1.
+The historically first definition of Cartan’s angular invariant in quaternionic hyperbolic geometry was
+given in [1]. In this paper, the authors defined the quaternionic Cartan angular invariant
+A(p) = A(p1, p2, p3)
+of an ordered triple p = (p1, p2, p3) of distinct points, pi ∈ Hn
+Q ∪ ∂Hn
+Q, to be the angle between the
+quaternion H(v1, v2, v3) and the real line R ⊂ Q, where vi is a lift of pi.
+They proved that A(p) does not depend on the chosen lifts, and it is the only invariant of a triple of
+isotropic points in the above sense.
+Next, we represent a convenient formula to compute A(p).
+Let p = (p1, p2, p3) be an ordered triple of distinct points, pi ∈ Hn
+Q ∪ ∂Hn
+Q, and v = (v1, v2, v3) be a lift
+of p = (p1, p2, p3). Then it follows from Proposition 1.1 and Lemma 2.2 that
+A∗(p) = arccos(−Re( H(v1, v2, v3))
+|H(v1, v2, v3)|
+)
+does not depend on the chosen lifts vi.
+Proposition 2.7 A(p) = A∗(p).
+Proof: The proof follows from an easy computation.
+The quaternionic Cartan angular invariant A(p) = A(p1, p2, p3) introduced above enjoys the following
+properties, see [1]:
+1. 0 ≤ A(p) ≤ π/2,
+2. If σ is a permutation, then
+A(pσ(1), pσ(2), pσ(3)) = A(p1, p2, p3),
+3. Let p = (p1, p2, p3) be an ordered triple of points in ∂Hn
+Q.
+Then p lies in the boundary of a
+quaternionic geodesic in Hn
+Q if and only if A(p) = π/2,
+4. Let p = (p1, p2, p3) be an ordered triple of points in ∂Hn
+Q. Then p lies in the boundary of a real plane
+in Hn
+Q if and only if A(p) = 0.
+10
+
+It is seen that this quaternionic Cartan angular invariant, in contrast to the Cartan angular invariant
+in complex hyperbolic geometry, is non-negative, symmetric, and one can show that it does not enjoy the
+cocycle property. We think that the definition of the Cartan angular invariant in quaternionic hyperbolic
+geometry given above does not explain well its relation with the classical Cartan angular invariant in
+complex hyperbolic geometry. In what follows, we discuss another possible definitions of quaternionic
+Cartan’s angular invariant and explain why the quaternionic Cartan angular invariant must be non-
+negative and symmetric.
+We start with the following simple fact which has a far reaching consequence for the construction of
+invariants of triples in quaternionic hyperbolic geometry. We think that this may also help for defining of
+invariants of triples in other hyperbolic geometries, for instance, in the hyperbolic octonionic plane.
+Proposition 2.8 Let p = (p1, p2, p3) be an ordered triple of distinct points, pi ∈ PQn. Then there exists
+a projective submanifold W ⊂ PQn of complex type of complex dimension 2 passing through the points pi,
+that is, pi ∈ W, i = 1, 2, 3. Moreover, this submanifold W can be chosen, up to the action of PU(n, 1; Q),
+to be the canonical complex submanifold PC2 ⊂ PQn.
+Proof: Let p = (p1, p2, p3) be an ordered triple of distinct points, pi ∈ PQn, and v = (v1, v2, v3) be a lift
+of p = (p1, p2, p3).
+First, let us suppose that pi and pj are not orthogonal, for all i ̸= j. Consider
+H(v1, v2, v3) = ⟨v1, v2, v3⟩ = ⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩.
+It follows from Corollary 2.3 and Lemma 2.2 that there exists λ1 such that H(v1λ1, v2, v3) is a complex
+number. Let us fix this λ1, and let w1 = v1λ1. Then it follows from Lemma 2.2 that H(w1, v2, v3) is a
+complex number for any lifts of p2 and p3. Let now λ2 = ⟨v2, w1⟩, w2 = v2λ2. We have that ⟨w1, w2⟩
+is real. Setting λ3 = ⟨v3, w1⟩ and w3 = v3λ3, we have that ⟨w3, w1⟩ is real. Since H(w1, w2, w3) ∈ C,
+it follows that ⟨w2, w3⟩ ∈ C. Therefore for this normalizarion all Hermitian products are complex. This
+implies that the complex span of w1, w2, w3 is C- unitary subspace of Qn,1 of dimension 3, see Section
+1.2.3, and, therefore, the points p1, p2, p3 lie in a projective submanifold W ⊂ PQn of complex type of
+complex dimension 2.
+Now let us suppose that the set {p1, p2, p3} contains orthogonal points. Assume, for example, that
+p1 and p2 are orthogonal, that is, ⟨v1, v2⟩ = 0, where v1 and v2 are lifts of p1 and p2.
+Let v3 be a
+lift of p3. Setting λ1 = ⟨v1, v3⟩, λ2 = ⟨v2, v3⟩, and w1 = v1λ1, w2 = v2λ2, w3 = v3, we have that all
+Hermitian products are real. It follows that the complex span of w1, w2, w3 is C- unitary subspace of Qn,1
+of dimension 3, and, therefore, the points p1, p2, p3 lie in a projective submanifold W ⊂ PQn of complex
+type of complex dimension 2. The rest follows from the results of Section 1.2.3.
+Corollary 2.4 Let p = (p1, p2, p3) be a triple of distinct negative points, pi ∈ Hn
+Q. Then p lies in a totally
+geodesic submanifold of Hn
+Q of complex type of complex dimension 2.
+Corollary 2.5 Let p = (p1, p2, p3) be a triple of distinct isotropic points, pi ∈ ∂Hn
+Q. Then p lies in the
+boundary of a totally geodesic submanifold of Hn
+Q of complex type of complex dimension 2.
+These results show that geometry of triples of points in PQn is, in fact, geometry of triples of points in
+PC2. Therefore, all the invariants of triples of points in PQn relative to the diagonal action of PU(n, 1; Q)
+can be constructed using 2-dimensional complex hyperbolic geometry.
+First, we give a new definition of the Cartan angular invariant in quaternionic hyperbolic geometry.
+Let p = (p1, p2, p3) be an ordered triple of distinct isotropic points, pi ∈ ∂Hn
+Q. By Corollary 2.5, we
+have that p lies in the boundary of a totally geodesic submanifold M of Hn
+Q of complex type of complex
+11
+
+dimension 2. We know that M is the projectivization of negative vectors in a 3-dimensional complex
+subspace V 3 of Cn,1, Cn,1 ⊂ Qn,1, and the boundary of H2
+C is the projectivization of isotropic vectors in
+V 3.
+Let v = (v1, v2, v3) be a lift of p = (p1, p2, p3). Then vi ∈ V 3. We define
+A∗∗ = A∗∗(p) = arg(−⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩).
+Now we briefly explain how to use this invariant to classify ordered triples of isotropic points relative
+to the diagonal action of PU(n, 1; Q).
+Let p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) be two ordered triples of distinct points in ∂Hn
+Q. Suppose that
+A∗∗(p) = A∗∗(p′). We will show that p and p′ are equivalent relative to the action of PU(n, 1; Q).
+By Corollary 2.5, we have that p is contained in the boundary of a totally geodesic submanifold M(p)
+of Hn
+Q of complex type of complex dimension 2. Also the same is true for p′, where p′ ∈ ∂M(p′). We
+know, see Section 1.2.3, that all such submanifolds are equivalent relative to the action of PU(n, 1; Q),
+therefore, there exists an element f ∈ PU(n, 1; Q) such that f(M(p)) = M(p′). So, we can assume without
+loss of generality that p and p′ are in the boundary of the same submanifold M = H2
+C ⊂ Hn
+Q. Then, by
+applying the classical result of Cartan, we have that there exists a complex hyperbolic isometry g of M,
+g ∈ PU(2, 1; C), such that g(p) = p′. This isometry g can be extended to an element of PU(n, 1; Q) by the
+Witt theorem. This proves that p and p′ are equivalent relative the action of PU(n, 1; Q).
+Next we show why it is more convenient to consider quaternionic Cartan’s angular invariant to be
+symmetric and non-negative (non-positive)
+We need the following lemma, see Lemma 7.1.7 in [13].
+Lemma 2.3 Let p = (p1, p2, p3) be an ordered triples of distinct points in ∂H2
+C. Then there exists a real
+plane P ⊂ H2
+C such that inversion (reflection) ip in P satisfies
+iP (p1) = p2,
+iP (p2) = p1,
+iP (p3) = p3.
+Remark 2.4 We recall that iP is an anti-holomorphic isometry of H2
+C and
+A(p2, p1p3) = −A(p1, p2, p3).
+Also, as it is easy to see that any anti-holomorphic isometry of H2
+C is a composition of an element of
+PU(2, 1; C) and an anti-holomorphic reflection.
+Proposition 2.9 Let p = (p1, p2, p3) be an ordered triples of distinct points in ∂Hn
+Q, n > 1. Then there
+exists an element f ∈ PU(n, 1; Q) such that
+f(p1) = p2,
+f(p2) = p1,
+f(p3) = p3.
+Proof: Repeating the arguments above, we can assume that p is in the boundary of the M = H2
+C ⊂ Hn
+Q.
+Let P ⊂ M = H2
+C be a real plane and iP be the reflection in P acting in M as in Lemma 2.3. We will
+show that this map can be extended to an isometry of Hn
+Q. Notice that iP as a map of M is not induced
+by a linear map, it is induced by a semilinear map, therefore, we cannot apply the Witt theorem in this
+case.
+Let K be a totally geodesic submanifold of Hn
+Q isometric to H2
+Q which contains M. An easy argument
+shows that there exists a totally geodesic submanifold N of complex type of complex dimension 2 in K
+intersecting M orthogonally along P. Let iN be the geodesic reflection in N. We have that iN is an
+12
+
+element of the isometry group of K, isomorphic to PU(2, 1; C), whose fixed point set is N. We notice that
+iN is induced by a linear map, therefore, it follows from the Witt theorem that iN can be extended to
+an isometry f in PU(n, 1; Q). Note that by construction f leaves M invariant and its restriction to M
+coincides with iP . Therefore, f(p1, p2, p3) = (p2, p1, p3). It is easy to see that the fixed point set of f in
+Hn
+Q is a totally geodesic submanifold of complex type of maximal dimension in Hn
+Q. This submanifold is
+globally equivalent to the canonical submanifold Hn
+C ⊂ Hn
+Q.
+Corollary 2.6 Let p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) be two ordered triples of distinct points in
+∂Hn
+Q.
+Suppose that A∗∗(p) = −A∗∗(p′).
+Then p and p′ are equivalent relative to the diagonal action
+of PU(n, 1; Q).
+Remark 2.5 This imply that if for two ordered triples of distinct isotropic points p = (p1, p2, p3) and
+p′ = (p′
+1, p′
+2, p′
+3) we have that |A∗∗(p)| = |A∗∗(p′)|, then p is equivalent to p′ relative to the diagonal action
+of group PU(n, 1; Q). Therefore, it is natural to consider instead of A∗∗ its absolute value. Then the
+invariant |A∗∗| lies in the interval [0, π/2] and is symmetric.
+Corollary 2.7 |A∗∗| = A∗.
+2.2.2
+Quaternionic Brehm’s invariants
+First, we recall the definition of the Brehm shape invariants in complex hyperbolic geometry.
+Let p = (p1, p2, p3) be an ordered triple of distinct points in Hn
+C and v = (v1, v2, v3) be a lift of
+p = (p1, p2, p3).
+Brehm [4] defined the invariant which he called the shape invariant, or σ - invariant:
+σ(p) = Re⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩
+⟨v1, v1⟩⟨v2, v2⟩⟨v3, v3⟩.
+It is easy to check that σ(p) is well- defined, that is, it does not depend on the chosen lifts, and it is
+invariant relative the diagonal action of the full isometry group of Hn
+C.
+We consider {p1, p2, p3} as the vertices of a triangle in hyperbolic space Hn
+C. Brehm [4] showed that
+the side lengths and the shape invariant are independent and characterize the triangle up to isometry.
+Now let p = (p1, p2, p3) be an ordered triple of distinct points in Hn
+Q, and v = (v1, v2, v3) be a lift of
+p = (p1, p2, p3).
+It is easy to check that if wi = viλi, then
+⟨w1, w2⟩⟨w2, w3⟩⟨w3, w1⟩(⟨w1, w1⟩⟨w2, w2⟩⟨w3, w3⟩)−1 =
+λ1
+|λ1|⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩(⟨v1, v1⟩⟨v2, v2⟩⟨v3, v3⟩)−1 λ1
+|λ1|.
+This formula and Proposition 1.1 imply that
+σ∗(p) = −Re(⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩[⟨v1, v1⟩⟨v2, v2⟩⟨v3, v3⟩]−1)
+is independent of the chosen lifts. Also, it is clear, that σ∗(p) is invariant relative to the diagonal action
+of PU(n, 1; Q).
+We call this number σ∗(p) the quaternionic σ-shape invariant.
+13
+
+Proposition 2.10 σ∗(p) ≥ 0.
+Proof: By applying Corollary 2.4, we can assume that p lies in M = H2
+C ⊂ Hn
+Q. Then the result follows
+from Proposition2.5.
+As the first application of the results above, we have the following.
+Theorem 2.1 A triangle in Hn
+Q is determined uniquely up to the action of PU(n, 1; Q) by its three side
+lengths and its quaternionic σ-shape invariant σ∗.
+Proof: Let p = (p1, p2, p3) be an ordered triple of distinct points in Hn
+Q. By applying Corollary 2.4, we
+may assume that p lies in M = H2
+C ⊂ Hn
+Q. Then the result follows from Proposition 2.3 and results in [4].
+In [5], Brehm and Et-Taoui introduced another invariant in complex hyperbolic geometry which they
+called the direct shape invariant, or, τ-invariant. Below, we recall the definition of this invariant.
+Let p = (p1, p2, p3) be an ordered triple of distinct points in Hn
+C and v = (v1, v2, v3) be a lift of
+p = (p1, p2, p3). Then the direct shape invariant is defined to be
+τ = τ(p) = H(v1, v2, v3)
+|H(v1, v2, v3)|.
+It is easy to check that τ(p) is independent of the chosen lifts. Also, it was proved in [5] that two trian-
+gles Hn
+C are equivalent relative to the diagonal action of PU(n, 1; C) if and only if the three corresponding
+edge lengths and the direct shape invariant τ of the two triangles coincide.
+Remark 2.6 Note that the σ-shape invariant is symmetric, but for the τ-shape invariant we have that
+τ(p2, p1, p3) = τ(p1, p2, p3). This implies that the σ-shape invariant (with the side lengths) describes trian-
+gles up to the full isometry group of complex hyperbolic space (which includes anti-holomorphic isometries),
+but τ-shape invariant (with the side lengths) describes triangles up to the group of holomorphic isometries
+PU(n, 1; C).
+Now we define an analogue of τ-shape invariant in quaternionic hyperbolic geometry. We start with
+the following lemma whose proof is based on a direct computation.
+Lemma 2.4 Let p = (p1, p2, p3) be an ordered triple of distinct points in Hn
+Q and v = (v1, v2, v3) be a lift
+of p = (p1, p2, p3). Let wi = viλi, λi ∈ Q, λi ̸= 0, then
+H(w1, w2, w3)|H(w1, w2, w3)|−1 =
+λ1
+|λ1|H(v1, v2, v3)|H(v1, v2, v3)|−1 λ1
+|λ1|.
+It is easy to see that H(w1, w2, w3)|H(w1, w2, w3)|−1 is similar to H(v1, v2, v3)|H(v1, v2, v3)|−1 for any
+λi ∈ Q, λi ̸= 0. Moreover, this similarity class contains a complex number, unique up to conjugation, see
+Proposition 1.1.
+Let τ ∗(p) denote a unique complex number with non-negative imaginary part in this similarity class.
+We define the quaternionic τ-shape invariant to be τ∗ = τ∗(p). It is clear that τ ∗(p) does not depend
+on the chosen lifts.
+14
+
+Proposition 2.11 Let p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) be two ordered triples of distinct points in Hn
+Q.
+Then these two triangles are equivalent relative to the diagonal action of PU(n, 1; Q) if and only if the
+three corresponding edge lengths and the quaternionic τ-shape invariant τ∗ of the two triangles coincide.
+Proof: By applying Corollary 2.4, we may assume that p and p′ are in M = H2
+C ⊂ Hn
+Q Then the result
+follows from Proposition 2.3.
+2.3
+Moduli of triples of positive points
+In this section, we describe the invariants associated to an ordered triple of positive points in PQn which
+define the equivalence class of the triple relative to the diagonal action of PU(n, 1; Q).
+First, we show how positive points in PQn are related to totally geodesic submanifolds in Hn
+Q isometric
+to Hn−1
+Q
+. We call such submanifolds of Hn
+Q totally geodesic quaternionic hyperplanes in Hn
+Q.
+We recall that π denotes a natural projection from Qn+1 \ {0} to the projective space PQn.
+If H ⊂ Hn
+Q is a totally geodesic quaternionic hyperplane, then H = π( ˜H) ∩ Hn
+Q, where ˜H ⊂ Qn,1 is
+a quaternionic linear hyperplane. Let ˜H⊥ denote the orthogonal complement of ˜H in Qn,1 with respect
+to the Hermitian form Φ. Then ˜H⊥ is a positive quaternionic line, and π( ˜H⊥) is a positive point in
+PQn. Thus, the totally geodesic quaternionic hyperplanes in Hn
+Q bijectively correspond to positive points.
+We call p = π( ˜H⊥) the polar point of a totally geodesic quaternionic hyperplane H. So, the invariants
+associated to an ordered triple of positive points in PQn are invariants of an ordered triple of totally
+geodesic quaternionic hyperplane in Hn
+Q.
+Let H1 and H2 be distinct totally geodesic quaternionic hyperplanes in Hn
+Q and p1, p2 be their polar
+points. Let v1 and v2 in Qn,1 be their lifts. Then we define
+d(H1, H2) = d(p1, p2) = ⟨v1, v2⟩⟨v2, v1⟩
+⟨v1, v1⟩⟨v2, v2⟩.
+It is easy to see that d(H1, H2) is independent of the chosen lifts of p1, p2, and that d(H1, H2) is invariant
+with respect to the diagonal action of PU(n, 1; Q).
+There is no accepted name for this invariant in the literature. It is not difficult to show using standard
+arguments that the distance or the angle between H1 and H2 is given in terms of d(H1, H2), (see, for
+instance, the Goldman [13] for the case of complex hyperbolic geometry). So, we will call this invariant
+d(H1, H2) the distance-angular invariant or, simply, d-invariant.
+Also, it is easy to see that:
+• H1 and H2 are concurrent if and only if d(H1, H2) < 1,
+• H1 and H2 are asymptotic if and only if d(H1, H2) = 1,
+• H1 and H2 are ultra-parallel if and only if d(H1, H2) > 1.
+Moreover, d(H1, H2) is the only invariant of an ordered pair of totally geodesic quaternionic hyperplanes
+in Hn
+Q. We have also that the angle θ between H1 and H2 (in the case d(H1, H2) < 1) is given by
+cos2(θ) = d(H1, H2),
+and the distance ρ between H1 and H2 (in the case d(H1, H2) ≥ 1) is given by
+cosh2(ρ) = d(H1, H2)
+15
+
+.
+We remark that 0 < θ ≤ π/2.
+We say that H1 and H2 are orthogonal if θ = π/2, this is equivalent to the equality d(H1, H2) = 0.
+Now let (H1, H2, H3) be an ordered triple of distinct totally geodesic quaternionic hyperplanes in Hn
+Q.
+Let p1, p2, p3 be the polar points of H1, H2, H3 and v1, v2, v3 be their lifts in Qn,1. Then, by applying
+Proposition 2.8, we can assume that p1, p2, p3 lie in a projective submanifold W ⊂ PQn of complex type
+of complex dimension 2 passing through the points pi. Moreover, this submanifold W can be chosen, up
+to the action of PU(n, 1; Q), to be the canonical complex submanifold PC2 ⊂ PQn. Therefore, we can
+assume without loss of generality that the coordinates of the vectors v1, v2, v3 are complex numbers.
+Let G = (gij) = (⟨vi, vj⟩) be the Gram matrix associated to the points p1, p2, p3 defined by the chosen
+vectors v1, v2, v3 as above. Then it follows from Proposition 2.2 and the proof of Proposition 2.8 that
+gii = 1, g1j = r1j ≥ 0, and g23 = r23eα. We call such a matrix G a complex normal form of the associated
+Gram matrix. Also, we call G the complex normalized Gram matrix.
+Next we construct the moduli space of ordered triples of distinct totally geodesic quaternionic hy-
+perplanes in Hn
+Q. We consider only the regular case, that is, when for all triples in question the spaces
+spanned by lifts of their polar points are regular, see Corollary 2.2. It is easy to see that in non-regular
+case, the totally geodesic quaternionic hyperplanes H1, H2, H3 are all asymptotic, that is, d(Hi, Hj) = 1,
+i ̸= j. It was shown in [10] that a similar problem, the congruence problem for triples of complex geodesic
+in complex hyperbolic plane, cannot be solved by using Hermitian invariants.
+An ordered triple H = (H1, H2, H3) of totally geodesic quaternionic hyperplanes in Hn
+Q is said to be
+generic if Hi and Hj are not orthogonal for all i, j = 1, 2, 3. It is clear that H is generic if and only if the
+corresponding triple of polar points is generic.
+We start with the following proposition:
+Proposition 2.12 Let H = (H1, H2, H3) and H′ = (H′
+1, H′
+2, H′
+3) be two ordered generic triples of dis-
+tinct totally geodesic quaternionic hyperplanes in Hn
+Q. Let p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) be their
+polar points. Let v = (v1, v2, v3) and v′ = (v′
+1, v′
+2, v′
+3) be their lifts in Qn,1 such that the Gram matrices
+G = (gij) = (⟨vi, vj⟩) and G′ = (g′
+ij) = (⟨v′
+i, v′
+j⟩) are complex normalized. Suppose that the spaces spanned
+by v1, v2, v3 and v′
+1, v′
+2, v′
+3 are regular. Then H and H′ are equivalent relative to the diagonal action of
+PU(n, 1; Q) if and only if G = G′ or ¯G = G′.
+Proof: If G = G′, then it follows from Corollary 2.2 that there exists a linear isometry L : Qn,1 −→ Qn,1
+such that L(vi) = v′
+i. Let us suppose that ¯G = G′. Then we have that g1j = g′
+1j because g1j and g′
+1j are
+real. Also, if g23 = r23eα, then g′
+23 = r23e−α. Let us consider the semi-linear map Lj : Qn,1 −→ Qn,1
+defined by the rule Lj(v) = jvj−1. Then we have that
+⟨Lj(vk), Lj(vl)⟩ = ⟨jvkj−1, jvlj−1⟩ = j−1⟨jvk, jvl⟩j−1 = // − ¯j⟨vk, vl⟩j−1 = j⟨vk, vl⟩j−1 = ⟨vk, vl⟩.
+Here we have used that jzj−1 = ¯z for any complex number z and that j−1 = −j.
+It follows that the Gram matrix of the vectors Lj(vk), k = 1, 2, 3, is equal to ¯G. Therefore, using the
+first part of the proof, we get that the triples v and v′ are equivalent relative to the diagonal action of
+U(n, 1; Q). This implies that H and H′ are equivalent relative to the diagonal action of PU(n, 1; Q).
+Let p = (p1, p2, p3) be an ordered generic triple of distinct positive points in PQn. Let v = (v1, v2, v3)
+be their lifts in Qn,1. Let H(v1, v2, v3) = (⟨v1, v2⟩⟨v2, v3⟩⟨v3, v1⟩) ∈ Q.
+Proposition 2.12 justifies the following definition.
+16
+
+We define the angular invariant A = A(p) of an ordered generic triple of distinct positive points
+p = (p1, p2, p3) in PQn to be the argument of the unique complex number b = b0 + b1 with b1 ≥ 0 in the
+similarity class of
+τ(v1, v2, v3) = H(v1, v2, v3)|H(v1, v2, v3)|−1.
+It follows from Lemma 2.4 and Corollary 1.1 that A = A(p) is defined uniquely by the real part of
+τ(v1, v2, v3) which does not depend on the chosen lifts v1, v2, v3.
+It is clear from the construction that A = A(p) is invariant with respect to the diagonal action of
+PU(n, 1; Q). Also, 0 ≤ A(p) ≤ π.
+By applying the proof of Proposition 2.8, we can chose lifts v = (v1, v2, v3) of p = (p1, p2, p3) such that
+the Gram matrix associated to p = (p1, p2, p3) defined by v = (v1, v2, v3) is the complex normalized Gram
+matrix of p = (p1, p2, p3), that is, gii = 1, g1j = r1j > 0, and g23 = r23eα, r23 >. It is clear that for these
+lifts A(p) = α.
+Now we are ready to describe the moduli space of ordered triples of distinct regular generic totally
+geodesic quaternionic hyperplanes in Hn
+Q.
+Theorem 2.2 Let H = (H1, H2, H3) and H′ = (H′
+1, H′
+2, H′
+3) be two ordered distinct regular generic
+totally geodesic quaternionic hyperplanes in Hn
+Q. Then H = (H1, H2, H3) and H′ = (H′
+1, H′
+2, H′
+3) are
+equivalent relative to the diagonal action of PU(n, 1; Q) if and only if d(Hi, Hj) = d(H′
+i, H′
+j) for all i < j,
+i, j = 1, 2, 3, and A(p) = A(p′), where p = (p1, p2, p3) and p = (p′
+1, p′
+2, p′
+3) are the polar points of H) and
+H′.
+Proof: We can choose lifts v = (v1, v2, v3) and v′ = (v′
+1, v′
+2, v′
+3) of p = (p1, p2, p3) and p = (p′
+1, p′
+2, p′
+3)
+such that the Gram matrices G and G′ associated to p and p′ defined by v and v′ are complex normalized.
+Then it follows from the definition of the Gram matrix that dij = d(pi, pj) = gijgji = |gij|2, and that
+A(p) = A(p1, p2, p3) = arg(g12 g23 g31) = arg(r12 g23j r31).
+The first equality implies that |gij| =
+�
+dij. Since H is generic, we have that r1j > 0 for all j > 1,
+and that g23 ̸= 0. Therefore, the second equality implies that A(p) = arg(g23). Thus, all the entries of
+the complex normalized Gram matrix G(p) of p are recovered uniquely in terms of the invariants dij and
+A(p) above. Now the proposition follows from Corollary 2.2.
+Next, as a corollary of Theorem 2.2, we give an explicit description of the moduli space of regular
+generic triples of totally geodesic quaternionic hyperplanes in Hn
+Q. First of all, it follows from Theorem 2.2
+that PU(n, 1; Q)-congruence class of an ordered regular generic triple H of distinct generic totally geodesic
+quaternionic hyperplanes in Hn
+Q complex is described uniquely by three d-invariants d12, d13, d23 and the
+angular invariant α = A(p1, p2, p3). Now, let G = (gij) be the complex normalized Gram matrix of H.
+Then gii = 1, g1j = r1j > 0, and arg(g23) = A(p1, p2, p3). We have that d1j = r2
+1j and d23 = r2
+23. Also,
+g23 = r23 eiα = r23(cos α + i sin α)
+.
+A straightforward computation shows that
+det G = 1 − (r2
+12 + r2
+13 + r2
+23) + 2r12r13r23 cos α.
+Using the lexicographic order, we define r1 =
+√
+d12, r2 =
+√
+d13, r3 =
+√
+d23.
+It follows from the Silvester criterium that det G ≤ 0, therefore, we have
+17
+
+Corollary 2.8 The moduli space M0(3) of regular generic totally geodesic quaternionic hyperplanes in
+Hn
+Q is homeomorphic to the set
+M0(3) = {(r1, r2, r3, α) ∈ R4 : ri > 0, α ∈ (o, π], 1 − (r2
+1 + r2
+1 + r2
+3) + 2r1r2r3 cos α ≤ 0}.
+Remark 2.7 If the triple H = (H1, H2, H3) is not generic, we need less invariants to describe the equiv-
+alence class of H. For instance, if H2 and H3 are both orthogonal to H1, we need only d(H2, H3).
+2.4
+Moduli of triples of points: mixed configurations
+As we know from Section 2.2.2, that, up to isometries of Hn
+Q, triples of points of Hn
+Q, that is, triangles,
+are characterized by the side lengths and the Brehm shape invariant. Also, triples of points in ∂Hn
+Q, that
+is, ideal triangles, are characterized by the Cartan angular invariant, see Section 2.2.1. The description of
+the moduli of triples of positive points was given in Section 2.3.
+In this section, we describe the invariants for mixtures of the three types of points in PQn relative to
+the diagonal action of PU(n, 1; Q). Using this, we describe the moduli of the corresponding configuration.
+First, we give a list of all the triples of points in PQn.
+Let p = (p1, p2, p3) be a triple of points in PQn. Then we have the following possible configurations
+(up to permutation).
+1. Ideal triangles: pi is isotropic for i = 1, 2, 3.
+2. Triangles: pi is negative, that is, pi in Hn
+Q for i = 1, 2, 3.
+3. Triangles of totally geodesic quaternionic hyperplanes in Hn
+Q.
+4. Triangles with two ideal vertices: p1 and p2 are isotropic, and (a) p3 is negative, (b) p3 is positive.
+5. Triangles with one ideal vertex: p1 is isotropic, and (a) p2, p3 are negative, (b) p2, p3 are positive,
+(c) p2 is negative, p3 is positive.
+6. Triangles with one negative vertex and two positive vertices: p1 is negative, and p2, p3 are positive.
+7. Triangles with one positive vertex and two negative vertices: p1 is positive, and p2, p3 are negative.
+The moduli of triangles in the first three cases have been already established in the previous sections
+Below, we describe invariants of mixed configurations 4-7.
+First, we recall the definition of so called η-invariant [13] in complex hyperbolic geometry.
+Let (p1, p2, q) be an ordered triple of points in PCn. We suppose that p1 and p2 are isotropic and q is
+positive. Let (v1, v2, w) be a lift of (p1, p2, p3). Then it is easy to check that the complex number
+η(v1, v2, w) = ⟨v1, w⟩⟨w, v2⟩
+⟨v1, v2⟩⟨w, w⟩
+is independent of the chosen lifts and will be denoted by η(p1, p2, q).
+We call the number η(p1, p2, q) the Goldman η-invariant. This invariant was introduced by Goldman
+[13] to study the intersections of bisectors in complex hyperbolic space. Later, Hakim and Sandler [15]
+generalized the Goldman construction for more general triples of points.
+The aim of this section is to define analogous invariants in quaternionic hyperbolic geometry and to
+prove the congruence theorems for all triples in question.
+18
+
+2.4.1
+Triangles with two ideal vertices
+Let p = (p1, p2, p3) be a triple of points in PQn.
+In what follows, we will denote by v the triple
+v = (v1, v2, v3, where vi is a lift of pi in Qn,1.
+First we consider the case when p1 and p2 are isotropic and p3 is positive. This configuration was
+considered by Goldman [13] in complex hyperbolic geometry. Geometrically it can be considered as a
+configuration of two points in the boundary of quaternionic hyperbolic space Hn
+Q and a totally geodesic
+quaternionic hyperplane in Hn
+Q.
+Let v1, v2 be isotropic vectors representing p1, p2 and v3 a positive vector representing p3. Consider
+the following quaternion:
+η(v1, v2, v3) = ⟨v1, v3⟩⟨v3, v2⟩⟨v1, v2⟩−1⟨v3, v3⟩−1.
+Now let us take another lifts of p = (p1, p2, p3): v′
+1 = v1λ1, v′
+2 = v2λ2, v′
+3 = v3λ3.
+It is easy to check that
+η(v1λ1, v2λ2, v3λ3) =
+¯λ1
+|λ1|η(v1, v2, w) λ1
+|λ1|.
+Therefore, this implies that η(v1, v2, v3) is independent of the choices of lifts of p2 and p3, and if we
+change a lift of p1, we get a similar quaternion.
+By applying Proposition 2.8, we can assume that p1, p2, p3 lie in a projective submanifold W ⊂ PQn
+of complex type of complex dimension 2 passing through the points pi. Moreover, this submanifold W
+can be chosen, up to the action of PU(n, 1; Q), to be the canonical complex submanifold PC2 ⊂ PQn.
+Therefore, we can assume without loss of generality that the coordinates of the vectors v1, v2, v3 are
+complex numbers.
+Let G =
+(gij) = (⟨vi, vj⟩) be the Gram matrix associated to the points p1, p2, p3. Then g11 = 0,
+g22 = 0. Since g33 = ⟨v3, v3⟩ > 0, replacing v3 by v3a, where a = 1/√g33, we may assume that g33 = 1.
+Replacing v2 by v2b, where b = 1/⟨v1, v2⟩, we may assume that g12 = 1. We keep the same notations
+for the modified vectors. After that, replacing v1 by v1b, where b = 1/⟨v3, v1⟩ and v2 by v2c, where
+c = ⟨v1, v3⟩, we get g12 = g13 = 1.
+So, after this re-scaling we get that G = (gij has the following
+entrances: g11 = g22 = 0, g12 = g13 = 1, g33 = 1, g23 is an arbitrary complex number.
+We call such a matrix G a complex normal form of Gram matrix associated to p = p1, p2, p3 . Also,
+we call G the complex normalized Gram matrix.
+It is clear that any vectors v1, v2 and v3 which represent p1, p2, and p3 generate a regular space in
+Qn,1. Therefore, repeating almost word for word the proof of Proposition 2.12, we get
+Proposition 2.13 Let p and p′ = (p′
+1, p′
+2, p′
+3) as above.
+Let v and v′ be their lifts such that the Gram matrices G = (gij) = (⟨vi, vj⟩) and G′ = (g′
+ij) = (⟨v′
+i, v′
+j⟩)
+associated to p and p′ are complex normalized.
+Then p and p′ are equivalent relative to the diagonal action of PU(n, 1; Q) if and only if G = G′ or
+G = G′.
+Again this justifies the following definition.
+Let p = (p1, p2, p3) be a triple of points as above. Then the quaternionic η-invariant, η = η(p), is
+defined to be the unique complex number b = b0 + b1 with b1 ≥ 0, in the similarity class of η(v1, v2, v3).
+19
+
+Theorem 2.3 Let p and p′ as above. Then p and p′ are equivalent relative to the diagonal action of
+PU(n, 1; Q) if and only if η(p) = η(p′).
+Proof: It follows from the above that we can choose lifts v = (v1, v2, v3) and v′ = (v′
+1, v′
+2, v′
+3) of
+p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) such that the Gram matrices G and G′ associated to p and p′
+defined by v and v′ are complex normalized. Then it follows from the definition of the Gram matrix that
+g23 = η(p). This implies that G = G′. Now the proof follows from Lemma 2.1.
+The case when p3 is negative is similar. The η-invariant is defined by the same formula and the proof of
+the congruence theorem is a slight modification of Theorem 2.3. Therefore, we have the following theorem.
+Theorem 2.4 Let p = (p1, p2, p3), where p1, p2 are isotropic and p3 is negative. Then the congruence
+class of p relative to the diagonal action of PU(n, 1; Q) is completely defined by the Goldman invariant
+η(p).
+We remark that this problem was considered by Cao, [7], see Theorem 1.1 item (ii).
+In order to
+describe the congruence class relative to the diagonal action of PU(n, 1; Q) of a triple p = (p1, p2, p3),
+where p1, p2 are isotropic and p3 is negative, he used the following invariants: the distance d between
+the unique quaternionic line L(p1, p2) passing through the points p1, p2 and the point p3, and the Cartan
+invariant A(p) of the triple p. Then he claimed that the congruence class relative to the diagonal action
+of PU(n, 1; Q) of p is completely defined by d and A(p). Below we give an example which shows that this
+claim is not correct.
+Example I. Let p = (p1, p2, p3), where p1, p2 are isotropic and p3 is negative. Let L be the unique
+quaternionic line passing through the points p1, p2. In our example, we consider the case when the point
+p3 is contained in L. Therefore, in this case, d = 0 and A(p) = π/2 for all triples satisfying our condition.
+Now we will show that among such triples there exist triples which are not congruent relative to the
+diagonal action of PU(n, 1; Q). Since the quaternionic line L is isometric to H4
+R, we have that there exists
+a totally geodesic submanifold P of L of real dimension 2 such that p1, p2 are in the ideal boundary of
+P and p3 ∈ P. In fact, P is a totally geodesic submanifold of Hn
+Q isometric to H1
+C. We consider P as a
+hyperbolic plane and p = (p1, p2, p3) as a triangle in P with two ideal vertices p1, p2 and one proper vertex
+p3. Let l be the unique geodesic in P defined by p1, p2. Let dl be the distance between l and p3. It is
+well-known from plane hyperbolic geometry, see [3], that a triangle p = (p1, p2, p3) is defined uniquely up
+to the isometry of P by dl. Therefore, fixing p1, p2 and moving p3 inside of P, we get an infinite family of
+non-congruent triangles relative to the diagonal action of PU(n, 1; Q) with the same invariants defined in
+[7].
+2.4.2
+Triangles with one ideal vertex
+Let p = (p1, p2, p3) be a triple of points in PQn. As usual, we will denote by v the triple v = (v1, v2, v3),
+where vi is a lift of pi in Qn,1.
+First, we consider a configuration when p1 is isotropic and p2, p3 are negative. So, in this case p1 and
+p2 and p3 represent a triangle in Hn
+Q with one ideal vertex.
+Let v2, v3 be negative vectors representing p2, p3 and v1 be an isotropic vector representing p1.
+Consider the following quaternion:
+η(v1, v2, v3) = ⟨v1, v3⟩⟨v3, v2⟩⟨v1, v2⟩−1⟨v3, v3⟩−1.
+It is easy to check that
+η(v1λ1, v2λ2, v3λ3) =
+20
+
+¯λ1
+|λ1|η(v1, v2, v3) λ1
+|λ1|.
+Therefore, this implies that η(v1, v2, v3) is independent of the choices of lifts of p2 and p3, and if we
+change a lift of p1, we get a similar quaternion.
+By applying Proposition 2.8, we can assume that p1, p2, p3 lie in a projective submanifold W ⊂ PQn
+of complex type of complex dimension 2 passing through the points pi. Moreover, this submanifold W
+can be chosen, up to the action of PU(n, 1; Q), to be the canonical complex submanifold PC2 ⊂ PQn.
+Therefore, we can assume without loss of generality that the coordinates of the vectors v1, v2, v3 are
+complex numbers.
+Let G =
+(gij) = (⟨vi, vj⟩) be the Gram matrix associated to the points p1, p2, p3. Then g11 = 0,
+g22 < 0, and g33 < 0. It is not difficult to show that by appropriate re-scaling we may assume that g11 = 0,
+g22 = −1, g33 = −1, g12 = 1, g23 = r23 > 0, and g13 is an arbitrary complex number.
+We call such a matrix G a complex normal form of Gram matrix associated to p1, p2, p3 . Also, we
+call G the complex normalized Gram matrix.
+It is clear that for p1, p2, p3 any vectors v1, v2 and v3 which represent p1, p2, and p3 generate a regular
+space in Qn,1. Therefore, repeating again almost word for word the proof of Proposition 2.12, we get
+Proposition 2.14 Let p and p as above.
+Let v and v′ be their lifts such that the Gram matrices
+G = (gij) = (⟨vi, vj⟩) and G′ = (g′
+ij) = (⟨v′
+i, v′
+j⟩) associated to p and p′ are complex normalized. Then p
+and p′ are equivalent relative to the diagonal action of PU(n, 1; Q) if and only if G = G′ or G = G′.
+Again this justifies the following definition.
+Let p = (p1, p2, p3) be a triple of points as above. Then the quaternionic η-invariant, η = η(p), is
+defined to be the unique complex number b = b0 + b1 with b1 ≥ 0, in the similarity class of η(v1, v2, v3).
+Theorem 2.5 Let p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) as above. Then p and p′ are equivalent relative to
+the diagonal action of PU(n, 1; Q) if and only if η(p) = η(p′) and d(p2, p3) = d(p′
+2, p′
+3).
+Proof: It follows from the above that we can choose lifts v = (v1, v2, v3) and v′ = (v′
+1, v′
+2, v′
+3) of
+p = (p1, p2, p3) and p = (p′
+1, p′
+2, p′
+3) such that the Gram matrices G and G′ associated to p and p′
+defined by v and v′ are complex normalized. Then it follows from the definition of the Gram matrix that
+η(p) = g13 and r23 =
+�
+d(p2, p3). This implies that G = G′. Now the proof follows from Lemma 2.1.
+It follows from this theorem that the congruence class relative to the diagonal action of PU(n, 1; Q) of
+p = (p1, p2, p3) is defined η(p) and the distance between p2 and p3.
+The case when p2, p3 are positive is similar. The η-invariant is defined by the same formula and the
+proof of the congruence theorem is a slight modification of Theorem 2.5. We only remark that geometrically
+this configuration is equivalent to one isotropic point and two totally geodesic quaternionic hyperplane in
+Hn
+Q.
+The case when p1 is isotropic and p2, p3 are negative was considered by Cao [7], see Theorem 1.1 item
+(iii). In order to describe the congruence class relative to the diagonal action of PU(n, 1; Q) of a triple
+p = (p1, p2, p3), where p1 is isotropic and p2, p3 are negative, he used the following invariants: the distance
+d1 between the unique quaternionic line L(p1, p2) passing through the points p1, p2 and p3, the distance d2
+between the unique quaternionic line L(p1, p3) passing through the points p1, p3 and p2, and the distance
+d3 between the points p2, p3. Then he claimed that the congruence class relative to the diagonal action of
+21
+
+PU(n, 1; Q) of a triple p is completely defined by d1, d2 and d3. Below we give an example which shows
+that this claim is not correct.
+Example II. This example is similar to Example I above. Let p = (p1, p2, p3), where p1 is isotropic
+and p2, p3 are negative. In our example, we consider the case when the points p1, p2, p3 are contained in
+a quaternionic line L. Therefore, in this case, d1 = 0 and d2 = 0 for all triples satisfying our condition.
+Now we will show that among such triples there exist triples which are not congruent relative to the
+diagonal action of PU(n, 1; Q). Since the quaternionic line L is isometric to H4
+R, we have that there exists
+a totally geodesic submanifold P of L of real dimension 2 such that p1 is in the ideal boundary of P, and
+p2, p3 ∈ P. As in the example above, we have that P is a totally geodesic submanifold of Hn
+Q isometric to
+H1
+C. We consider P as a hyperbolic plane and p = (p1, p2, p3) as a triangle in P with one ideal vertex p1
+and two proper vertices p2, p3. It is well-known from plane hyperbolic geometry, see [3], that a triangle
+p = (p1, p2, p3) is defined uniquely up to the isometry of P by d3 and by its angles at the vertices p2, p3.
+Therefore, fixing p2, p3 and moving p1 along the ideal boundary of P, we get an infinite family of triangles
+with the same d3 and with different angles at the vertices p2, p3. This implies that there exist an infinite
+family of non-congruent triangles relative to the diagonal action of PU(n, 1; Q) with the same invariants
+defined in [7].
+2.4.3
+Triangles with one negative vertex and two positive vertices
+Let p = (p1, p2, p3) be a triple of points in PQn. In this section, we consider a configuration when p1 is
+negative, and p2, p3 are positive. So, in this case p1 represents a point in Hn
+Q, and p2 and p3 represent two
+totally geodesic quaternionic hyperplane H2 and H3 in Hn
+Q.
+Let v1 a negative vector representing p1 and v2, v3 be positive vectors representing p2, p3.
+Let η(v1, v2, v3) be following quaternion:
+η(v1, v2, v3) = ⟨v1, v3⟩⟨v3, v2⟩⟨v1, v2⟩−1⟨v3, v3⟩−1.
+We see that η(v1, v2, v3) is not well-defined when the points p1 and p2 are orthogonal, that is,
+⟨v1, v2⟩ = 0.
+It follow from the definition of polar points that p1 is orthogonal to p2 if and only if
+p1 ∈ H2. Also we see that η(v1, v2, v3) = 0 when p1 is orthogonal to p3 or p2 is orthogonal to p3. We
+analyze all these special configurations in the end of this section and show that in all these cases we need
+less invariants to describe the congruence class then in generic case.
+We say that a triple p = (p1, p2, p3) as above is generic if and only if all the pairs (pi, pj) are not
+orthogonal, i ̸= j.
+In what follows, let p = (p1, p2, p3) be a generic triple as above. It is easy to check that
+η(v1λ1, v2λ2, v3λ3) =
+¯λ1
+|λ1|η(v1, v2, v3) λ1
+|λ1|.
+Therefore, this implies that η(v1, v2, v3) is independent of the choices of lifts of p2 and p3, and if we change
+a lift of p1, we get a similar quaternion.
+By applying Proposition 2.8 again, we can assume that p1, p2, p3 lie in a projective submanifold
+W ⊂ PQn of complex type of complex dimension 2. Moreover, this submanifold W can be chosen, up
+to the action of PU(n, 1; Q), to be the canonical complex submanifold PC2 ⊂ PQn. Therefore, we can
+assume without loss of generality that the coordinates of the vectors v1, v2, v3 are complex numbers.
+Let G = (gij) = (⟨vi, vj⟩) be the Gram matrix associated to the triple of points (p1, p2, p3). Then
+g11 < 0, g22 > 0, and g33 > 0. It is not difficult to show that by appropriate re-scaling we may assume
+that g11 = −1, g22 = 1, g33 = 1, g12 = r12 > 0, g13 = r13 > 0, and g23 is an arbitrary complex number.
+22
+
+We call such a matrix G a complex normal form of Gram matrix associated to (p1, p2, p3). Also, we
+call G the complex normalized Gram matrix.
+Let p = (p1, p2, p3) be a triple of points as above. Then the quaternionic η-invariant, η = η(p), is
+defined to be the unique complex number b = b0 + b1 with b1 ≥ 0, in the similarity class of η(v1, v2, v3).
+As before we have
+Proposition 2.15 Let p and p′) as above.
+Let v and v′ be their lifts such that the Gram matrices
+G = (gij) = (⟨vi, vj⟩) and G′ = (g′
+ij) = (⟨v′
+i, v′
+j⟩) associated to p and p′ are complex normalized. Then p
+and p′ are equivalent relative to the diagonal action of PU(n, 1; Q) if and only if G = G′ or G = G′.
+Again this justifies the following definition.
+Let p = (p1, p2, p3) be a triple of points as above. Then the quaternionic η-invariant, η = η(p), is
+defined to be the unique complex number b = b0 + b1 with b1 ≥ 0,
+Now we define one more invariant. Let q1 be a negative point and q2 be a positive point. Let w1 be a
+vector representing q1 and w2 be a vector representing q2. Then we define
+d(q1, q2) = d(w1, w2) = ⟨w1, w2⟩⟨w2, w1⟩
+⟨w1, w1⟩⟨w2, w2⟩.
+It is easy to see that d(q1, q2) is independent of the chosen lifts w1, w2, and that d(q1, q2) is invariant
+with respect to the diagonal action of PU(n, 1; Q).
+We call d(q1, q2) the mixed distant invariant associated to the points q1, q2. By applying the standard
+arguments it is easy to prove that this invariant defines the distance ρ between the point q1 ∈ Hn
+Q and the
+totally geodesic quaternionic hyperplane H in Hn
+Q whose polar point is q2, namely,
+cosh2(ρ(q1, H)) = −d(q1, q2).
+Theorem 2.6 Let p = (p1, p2, p3) and p′ = (p′
+1, p′
+2, p′
+3) as above.
+Then p and p′ are equivalent rel-
+ative to the diagonal action of PU(n, 1; Q) if and only if η(p) = η(p′), d(p1, p2) = d(p′
+1, p′
+2), and
+d(p1, p3) = d(p′
+1, p′
+3).
+Proof: It follows from the above that we can choose lifts v = (v1, v2, v3) and v′ = (v′
+1, v′
+2, v′
+3) of
+p = (p1, p2, p3) and p = (p′
+1, p′
+2, p′
+3) such that the Gram matrices G and G′ associated to p and p′
+defined by v and v′ are complex normalized. Then it follows from the definition of the Gram matrix that
+r12 =
+�
+d(p1, p2), r13 =
+�
+d(p1, p3), and g23 = η(p)(r12/r13). This implies that G = G′. Now the proof
+follows from Lemma 2.1.
+As a corollary of this theorem we have
+Theorem 2.7 Let p1 ∈ Hn
+Q and let H1, H2 be two totally geodesic quaternionic hyperplane in Hn
+Q whose
+polar points are p2 and p3, respectively. Then the congruence class of the triple (p1, H1, H2) relative to the
+diagonal action of PU(n, 1; Q) is defined uniquely by two distances ρ(p1, H1), ρ(p1, H2), the d-invariant
+d(H1, H2), and the angular invariant of the triple (p1, p2, p3).
+Now we consider some special configurations of triples p = (p1, p2, p3). Let, for instance, p be a config-
+uration where all the pairs (pi, pj), i ̸= j, are orthogonal. In this case, the totally geodesic quaternionic
+hyperplanes H1 and H2 with polar points p2 and p3 intersect orthogonally and p1 lies in their intersection.
+It is clear then that all such configurations are congruent relative to the diagonal action of PU(n, 1; Q).
+23
+
+So, the moduli space in this case is trivial. Another example: suppose that p1 is orthogonal to p2, and p1
+is orthogonal to p3. This implies that H1 and H2 intersect, and p1 lies in their intersection. This implies
+that in order to describe the congruence class of such configuration we need only the d-invariant of H1
+and H2, the angle between H1 and H2. Another special configuration may be analyzed easily in a similar
+way.
+2.4.4
+Triangles with one positive vertex and two negative vertices
+In this section, we consider a triple (p1, p2, p3), where p1 is positive and p2, p3 are negative.
+Geometrically, this corresponds to a totally geodesic quaternionic hyperplane H in Hn
+Q and two points
+p2 and p3 in Hn
+Q.
+Theorem 2.8 Let H be a totally geodesic quaternionic hyperplane H in Hn
+Q with polar point p1, and p2
+and p3 in Hn
+Q. Suppose that p = (p1, p2, p3) is generic. Then the congruence class of the triple (H, p1, p2)
+is defined uniquely by two distances ρ(p1, H), ρ(p1, H), and η(p) = η(p1, p2, p3).
+Proof: The proof of this theorem is slight modification of the proof of Theorem 2.6.
+References
+[1] B.N. Apanasov and I. Kim, Cartan angular invariant and deformations of rank 1 symmetric spaces.
+Sb. Math. 198 (2007), 147-169.
+[2] E. Artin, Geometric algebra, Interscience Publishers, Inc., New York, 1957.
+[3] A.F. Berdon, The geometry of Discrete Groups. Springer- Verlag, Berlin, 1983. xx+304 pp.
+[4] U. Brehm, The shape invariant of triangles and trigonometry in two-point homogeneous spaces.
+Geom. Dedicata 33 (1990), no. 1, 59–76.
+[5] U. Brehm, B. Et-Taoui, Congruence criteria for finite subsets of complex projective and complex
+hyperbolic spaces. Manuscripta Math. 96 (1998), no. 1, 81–95.
+[6] J.L. Brenner, Matices of quaternions. Pacific J. Math. 1 (1950), 329–335.
+[7] W.S. Cao, Congruence classes of points in quaternionic hyperbolic space. Geom. Dedicata 180
+(2016), 203-228.
+[8] E. Cartan, Sur le groupe de la g´eom´etrie hypersph´erique. Comment. Math. Helv. 4, 158-171 (1932)
+[9] S.S. Chen and L. Greenberg, Hyperbolic spaces. In Contributions to analysis(a collection of papers
+dedicated to Lipman Bers), Academic Press, New York (1974), 49–87.
+[10] H. Cunha, F. Dutenhener, N. Gusevskii, and R. Thebaldi, The moduli space of complex geodesics
+in the complex hyperbolic plane. J. Geom. Anal. 22 (2012), no. 2, 295–319.
+[11] H. Cunha, N. Gusevskii, On the moduli space of quadruples of points in the boundary of complex
+hyperbolic space. Transform. Groups. (2010), no.2, 261-283.
+[12] H. Cunha, N. Gusevskii, The moduli space of points in the boundary of complex hyperbolic space.
+J. Geom. Anal. 22 (2012), no. 2, 1-11.
+24
+
+[13] W.M. Goldman, Complex hyperbolic geometry. Oxford Mathematical Monographs. Oxford
+Science Publications. The Clarendon Press, Oxford University Press, New York, 1999. xx+316
+pp.
+[14] R. Hofer, m-point invariants of real geometries. Beitr¨age Algebra Geom. 40 (1999). no. 1, 261-266.
+[15] J. Hakim, H. Sandler, Applications of Bruhat Decompositions to Complex Hyperbolic Geometry.
+J. Geom. Anal. 10 (2000), no.3, 435-453.
+[16] J. Hakim, H. Sandler, Standard position for objects in hyperbolic space. J. Geom. 68 (2000),
+100-113.
+[17] W. Scharlau, Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften
+[Fundamental Principles of Mathematical Sciences], 270. Springer-Verlag, Berlin, 1985. xx+421
+pp. pp.
+25
+

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+Combining Self-labeling with Selective Sampling
+Jedrzej Kozal1 , Michał Wo´zniak1
+1Department of Systems and Computer Networks
+Wrocław University of Science and Technology, Wrocław, Poland
+{jedrzej.kozal, michal.wozniak}@pwr.edu.pl
+Abstract
+Since data is the fuel that drives machine learn-
+ing models, and access to labeled data is gener-
+ally expensive, semi-supervised methods are con-
+stantly popular.
+They enable the acquisition of
+large datasets without the need for too many ex-
+pert labels. This work combines self-labeling tech-
+niques with active learning in a selective sampling
+scenario.
+We propose a new method that builds
+an ensemble classifier. Based on an evaluation of
+the inconsistency of the decisions of the individual
+base classifiers for a given observation, a decision
+is made on whether to request a new label or use the
+self-labeling. In preliminary studies, we show that
+na¨ıve application of self-labeling can harm perfor-
+mance by introducing bias towards selected classes
+and consequently lead to skewed class distribu-
+tion. Hence, we also propose mechanisms to reduce
+this phenomenon. Experimental evaluation shows
+that the proposed method matches current selective
+sampling methods or achieves better results.
+1
+Introduction
+Active learning [Cohn et al., 1994] is the area of machine
+learning where a training set is constructed by selecting the
+most informative samples that can speed up training. New
+labeled learning examples are obtained by queering, i.e., re-
+questing ground truth labels from an oracle. To create a query,
+we use a model trained with a small number of labeled sam-
+ples. Stream-Based Selective Sampling [Cohn et al., 1994] is
+based on the assumption that acquiring new unlabeled train-
+ing examples is relatively inexpensive. We process a single
+sample at a time, and decide whether it should be labeled by
+oracle or discarded. In this work, we propose a new method
+that combines self-labeling with active learning in Stream-
+Based Selective Sampling scenario.
+An overview of our method is provided in Fig. 1. We hope
+that this approach could allow for the cost-efficient creation of
+bigger labeled datasets. Self-labeling could introduce noisy
+labels into the dataset [Han et al., 2019], as in most cases,
+models have non-zero classification error. In [Algan and Ulu-
+soy, 2020] various types of noises and their impact on deep
+Unlabeled data
+model
+prediction
+. . .
+. . .
+confidence and
+consistency check
+FAILED
+PASSED
+request
+label
+bootstrapped
+training
+prior
+filter
+bootstrapped
+training
+ensemble
+predictions
+Training with label from annotator
+(Active Learning)
+Training with predicted label
+(Self-labeling)
+Figure 1: Overview of the proposed method. We utilize ensemble
+predictions to determine whether a given sample could be added to
+the dataset with the predicted label (self-labeling) or should be la-
+beled by oracle (active learning). More specifically, we check if
+obtained support exceeds a predefined threshold and if all confident
+predictions return to the same class. If not, we check if the budget,
+create a query, and train with bootstrapping. Otherwise, we filter out
+and drop the samples from the current majority class (prior filter),
+and perform bootstrapped training with label obtained from predic-
+tion.
+learning performance were analyzed. It was found for vari-
+ous noise types that, with the increase of the noise ratio, test
+accuracy decreased. and with the increase of dataset size,
+test accuracy increases. In self-labeling, errors made by the
+classifier introduce wrong labels to the dataset, but as we la-
+bel new samples, the overall data volume increases. If the
+gain in accuracy from increasing the dataset size surpasses
+the performance loss from the wrong labels, we can use self-
+labeling to boost classification performance. In this work, we
+utilize this dynamic to improve classification performance.
+The main contributions of this work are following:
+• An analysis of the problems, that arise when we apply
+self-labeling to an active learning scenario
+• New method was proposed based on classifier commit-
+tee, that integrates self-labeling to active learning
+• Thorough experimental evaluation with multiple dataset
+and settings
+arXiv:2301.04420v1  [cs.LG]  11 Jan 2023
+
+2
+Related Works
+2.1
+Active learning
+The most popular active learning strategy is based on uncer-
+tainty sampling [Lewis and Gale, 1994], where model sup-
+ports are utilized as an information source about learning
+example usefulness. Fragment of the sample space, where
+the support for the samples is low, is called the region of
+uncertainty [Atlas et al., 1989]. This concept was used in
+[Lewis and Catlett, 1994] to select samples for labeling with
+the lowest difference in computed support and the predefined
+threshold. In [Zliobaite et al., 2014], authors proposed un-
+certainty sampling with a variable threshold for data stream
+mining. In margin sampling [Scheffer et al., 2001], queries
+are created by selecting samples with the smallest difference
+in probabilities of two classes with the largest confidence. It
+was shown in [Ramirez-Loaiza et al., 2017] that this algo-
+rithm performs on par with more computationally expensive
+ensemble-based methods. In [Jiang and Gupta, 2019], mod-
+ification of standard margin sampling was proposed, were
+samples are selected based on the smallest classification mar-
+gin of all models in the ensemble.
+Query by Committee
+algorithms measures disagreement between members of an
+ensemble to choose the most informative samples. In vote
+entropy, [Argamon-Engelson and Dagan, 1999] samples are
+selected based on the entropy of ensemble vote distribution.
+A modified version of this algorithm [G´omez-Ca˜n´on et al.,
+2021] select samples with the highest average predictions en-
+tropy. Another possible disagreement measure is maximum
+disagreement sampling [McCallum and Nigam, 1998], where
+KL divergence is used.
+2.2
+Self-labeling
+Self-supervised learning [Jaiswal et al., 2020] aims at learn-
+ing valuable data representation without annotation. To ob-
+tain good representations, we need to define some pretext
+tasks for a model to solve. The first attempts of creating self-
+supervised learning involved auto-encoders [Bengio et al.,
+2006], patch location prediction [Doersch et al., 2015], in-
+painting [Pathak et al., 2016], or rotation prediction [Gidaris
+et al., 2018]. Clustering was utilized as a pretext task [Caron
+et al., 2018; Asano et al., 2019] for training deep data repre-
+sentations. Another research direction is contrastive learning
+[Chen et al., 2020], where the pretext task is based on learn-
+ing close representations for the same sample with different
+augmentations applied and distant representations for dissim-
+ilar samples. Pseudo-labels are also used for semi-supervised
+learning [Rizve et al., 2021]. Authors of [Lee, 2013] uti-
+lize the outputs of the classifier with the highest confidence
+as a target for unlabeled data. In [Sohn et al., 2020], new
+methods were proposed that utilize high-confidence pseudo-
+labels generated with weakly-augmented images. Next, these
+pseudo-labels are used as a target for the strongly-augmented
+version of the same image.
+2.3
+Active learning with Self-labeling
+In [Wang et al., 2017] combination of automatic pseudo-
+labeling and active learning was proposed for a pool-based
+setting. Authors found that utilization of pseudo-labels can
+improve the labeling efficiency of active learning algorithms,
+and error rates of automatically assigned labels are low for
+the convolutional neural network. Authors of [Sim´eoni et al.,
+2019] introduce a new method that combines semi-supervised
+learning with pseudo-labels and active learning.
+Korycki
+and Krawczyk [Korycki and Krawczyk, 2021] have com-
+bined self-labeling with active learning for learning from data
+streams.
+3
+Method
+In this section, we describe the setting and introduce our
+method. We also provide results of preliminary experiments
+with the dynamic imbalance.
+3.1
+Selective sampling
+First, we introduce the selective sampling framework that our
+work is based on. We assume an access to small set of la-
+beled data L = {(xm, ym)}M
+m=1 and stream of unlabeled data
+U = {(xn}N
+n=1 with x ∈ X, y ∈ Y, where X and Y are the
+input space and the set of labels respectively. Our goal is to
+train a model f, that predicts the support for input sample x,
+namely: p(y|x) = fθ(x), where θ is set of model parameters.
+Final model prediction is given by ˆy = arg maxi p(yi|x).
+We denote maximum support for sample x as maxi p(yi|x).
+The general algorithm for selective sampling is provided in
+appendix A. We assume the same cost of obtaining label from
+an oracle for each sample in U. For this reason, we define
+budget B as the number of samples that can be labeled, with
+the exception of presenting results when we refer to budget
+as the fraction of all samples that can be labeled.
+3.2
+Informativeness computation
+We employ differences in supports obtained from different
+models in an ensemble as a source of informativeness. First,
+the ensemble of L base classifiers is trained. For the unla-
+beled sample x each model l in a committee computes sup-
+ports pl(y|x). Next, we check if at least half of the classifiers
+in the ensemble provided supports that exceed a predefined
+support threshold τ.
+�
+l
+1maxc pl(yc|x)>τ > L
+2
+(1)
+If more than half of the models return confident predictions
+and these models output the same prediction, we add (x, ˆy) to
+L. Otherwise, we query an oracle with x. By choosing sam-
+ples with consistent, highly confident predictions, we avoid
+assigning the wrong label to a sample. From an active learn-
+ing perspective, these learning examples are not valuable, as
+models already return confident predictions for them. How-
+ever, we hope that by a faster increase in the number of la-
+beled samples, we can obtain improvements in classification
+accuracy.
+3.3
+Bootstrapped training
+We train initial models with bootstrapping of labeled part of
+data L. This corresponds to sampling number of repeats of
+each sample from the Poisson distribution with λ = 1. This
+part of our method is inspired by Online Bagging [Oza and
+
+Russell, 2001] method, introduced for data stream classifica-
+tion. During training with an unlabeled stream, we use boot-
+strapping for new learning examples added to the dataset. In
+the case of training with ground truth label from oracle, we
+use λ = 1. When updating the dataset with a sample labeled
+based on model prediction, we calculate λ as:
+λ = maxl,c pl(yc|x)
+τ
+− 1B=0
+(2)
+where τ is the same threshold used earlier for selecting confi-
+dent predictions. When B > 0, then λ is always greater than
+one. As a result, samples labeled based on model prediction
+will be more frequently added to the dataset than learning
+examples from initial dataset L. To avoid the negative influ-
+ence of incorrect model predictions after the budget ended we
+change lambda calculation after labeling budget was spent. In
+such case λ < 1, assuming that value of τ is not significantly
+lower than p. Consequently, updates to datasets are still per-
+formed, while the negative impact of incorrect predictions is
+limited. Values of λ for each sample are stored in λ vector.
+Upon an update, we generate separate datasets by bootstrap-
+ping. The number of repeats of a single sample in a dataset
+is limited to 4. The ensemble training procedure for the pro-
+posed method is given in algorithm 1.
+Algorithm 1 Bootstrapped training
+Require: L - set of labeled data with M elements, {fθ}L
+- ensemble of L models, λ - vector with parameters for
+Poisson distribution for each sample in L
+1: for l ∈ {0, L} do
+2:
+r ∼ Pois(λ)
+3:
+r ← min(r, 4)
+4:
+D ← ∅
+5:
+for i ∈ {0, M} do
+6:
+(x, y) ← Li
+7:
+for j ∈ {0, ri} do
+8:
+D ← D ∪ {(x, y)}
+9:
+end for
+10:
+end for
+11:
+train fθl with D
+12: end for
+3.4
+Dynamic Imbalance
+Na¨ıve usage of self-labeling in selective sampling can intro-
+duce imbalance in the training set L. To demonstrate this,
+we conduct a preliminary experiment with synthetic data. We
+generate simple datasets by sampling from 2D Gaussian dis-
+tribution for easier visualization. We study two scenarios that
+may occur in practice. In the first case, the dataset consists of
+three balanced classes, but one of the classes is easier to learn
+than the rest. In the second scenario dataset with two classes
+is imbalanced.
+We sample 300 learning examples, plotted in Fig. 2 on the
+left-hand side. Datasets are used for initial training of Multi-
+layer Perceptron with 5 neuron single hidden layer. Next,
+we generate a stream with 3000 learning examples, sample
+data from the stream, and obtain model predictions. When
+model confidence exceeds 0.95, we expand the training set
+with learning examples and predicted labels. For simplicity,
+we do not use bootstrapping in this experiment. When model
+confidence is below 0.7, we create a query to obtain a ground-
+truth label. Changes in the percentage of labels in training set
+during training with an unlabeled data stream are presented
+in Fig. 2 in the middle.
+In the first scenario, over time percentage of samples la-
+beled as the third class grows until it utilizes approximately
+40% of all data. This result shows that na¨ıve utilization of
+self-labeling can disturb class distribution, even if the original
+data is balanced. In the second scenario, the initial imbalance
+ratio is 1:4, however, after approximately 800 iterations, it is
+closer to 1:5. This shows that initial bias in the data distri-
+bution can be strengthened by self-labeling. Please note that
+in this experiment algorithm have access to the ground truth
+labels by creating queries for samples with low confidence,
+and yet, the class distribution change over time.
+3.5
+Prior filter
+To address the issue of dynamic imbalance, we introduce a
+method that prevents training when the current prior estima-
+tion for the predicted class is too high. We use the last k labels
+from L and compute the percentage of samples that have the
+same label as the predicted class:
+ˆp = 1
+k
+M
+�
+i=M−k
+1yi=ˆy
+(3)
+This value can be interpreted as an estimation of the cur-
+rent class prior. Only the last k labels were used because,
+as shown in preliminary experiments, class distribution can
+change over time.
+We compute difference between ˆp and
+prior of perfectly balanced dataset:
+∆p = ˆp − 1
+C
+(4)
+Where C is the number of all classes. When ∆p > 0 we
+disallow training. We do not apply this prior filter to labels
+obtained from an oracle. A similar approach was proposed
+earlier in [Komorniczak et al., 2022] in the context of the
+data stream processing, however it estimated prior with re-
+gression models and switching labels for the majority class.
+Here we estimate prior directly from model predictions and
+skip samples from majority classes, which is similar to un-
+dersampling.
+We repeat previous preliminary experiments with the prior
+filter applied. We use k = 50 last samples. Results are plotted
+in Fig. 2 on the right-hand side. The proposed method can
+keep class distribution balanced in the first setting and, over
+time, improve initial class distribution in the second setting.
+3.6
+Self-labeling selective sampling
+The complete algorithm for Self-labeling selective sampling
+(SL2S) along with time complexity analysis is provided in
+appendix B.
+
+Figure 2: Dynamic imbalance of classes when applying self-labeling directly during selective sampling. We consider two settings: in the
+first three classes with balanced prior distribution, but a single class is easier to learn than others (up), and the second imbalanced binary
+classification problem (bottom). Generated 2-D datasets are plotted on the left-hand side. When applying self-labeling directly (middle), we
+observe a change in the class distribution in the training set. This problem can be avoided when we apply dynamic balancing (right).
+4
+Experimental Setup
+This section provides a detailed description of the methods,
+datasets, and tools used to conduct experiments.
+4.1
+Datasets
+We utilize datasets from the UCI repository [Dua and Graff,
+2017] with a wide range of datasets with different sizes, num-
+ber of classes, number of attributes, and imbalance ratio (IR).
+The detailed information about data used in experiments is
+presented in Tab. 1. The complete list of features and proce-
+dures for loading data are provided in appendix C.
+Table 1: Datasets used for experiments. IR was computed by taking
+a ratio of class with the highest and lowest number of samples.
+dataset name
+size
+#class
+#attributes
+IR
+adult [Kohavi, 1996]
+48842
+2
+14
+3.1527
+bank marketing [Moro et al., 2014]
+45211
+2
+17
+7.5475
+firewall [Ertam and Kaya, 2018]
+65478
+3
+12
+2.9290
+chess [Dua and Graff, 2017]
+20902
+15
+40
+22.919
+nursery [M. Olave, 1989]
+12958
+4
+8
+13.1707
+mushroom [Dua and Graff, 2017]
+8124
+2
+22
+1.0746
+wine [Cortez et al., 2009]
+4873
+5
+12
+13.5082
+abalone [Nash et al., 1994]
+4098
+11
+8
+21.5417
+4.2
+Metrics and evaluation
+Due to high values of IR for some datasets, we decided to em-
+ploy balanced accuracy [Brodersen et al., 2010] as primary
+performance metrics for our experiments. In our evaluation,
+we focus on the impact of budget size and seed size used for
+training of the initial model, as these two factors can impact
+the results the most. All values of metrics reported in this pa-
+per were obtained with a separate test set. Code was imple-
+mented in Python with the utilization of scikit-learn library
+[Pedregosa et al., 2011]. The codebase with the method and
+experiment implementations are available on github 1.
+4.3
+Baselines
+To perform fair evaluation, we compare the proposed method
+to commonly used algorithms in selective sampling literature:
+• random - random selection of samples for query
+• fixed uncertainty [Lewis and Catlett, 1994] - selection of
+samples based on a static confidence threshold
+• variable uncertainty [Zliobaite et al., 2014] - modifica-
+tion of fixed uncertainty that adjust confidence threshold
+based on the current size of the uncertainty region
+• classification margin [Scheffer et al., 2001] - a method
+that computes the difference in confidence between
+classes with two biggest supports
+• vote entropy [Argamon-Engelson and Dagan, 1999] -
+queries are based on ensemble vote entropy
+• consensus entropy [G´omez-Ca˜n´on et al., 2021] - sam-
+ples are selected based on the highest average prediction
+entropy
+• max disagreement [McCallum and Nigam, 1998] - com-
+putes KL-divergence between output class distribution
+and consensus distribution
+• min margin [Jiang and Gupta, 2019] - a method that se-
+lects samples based on minimum classification margin
+for all models in the ensemble
+In the case of methods that were created for the pool-based
+scenario, we adapt them by introducing the informativeness
+1https://github.com/w4k2/active-learning-data-streams
+
+1.0
+1.0
+class 1
+4
+class 2
+0.8
+0.8
+class 3
+percentage
+class percentage
+2
+0.6 
+0.6 
+0
+0.4
+0.4
+class
+-2
+0.2
+0.2 
+-4
+0.0
+0.0
+-5
+0
+5
+0
+500
+1000
+1500
+2000
+0
+200
+400
+600
+800
+iterations
+iterations
+1.0
+1.0
+4
+0.8
+0.8
+2
+percentage
+ percentage
+0.6
+0.6
+0
+class
+0.4
+class E
+0.4
+.
+-2
+class 1
+0.2
+0.2
+-4
+class 2
+0.0
+0.0
+-4
+-2
+2
+o
+500
+1000
+1500
+2000
+0
+200
+400
+600
+iterations
+iterationsthreshold. For each committee-based method, we use 9 base
+classifiers and employ bootstrapping during initial training.
+All methods were trained with Multi-layer Perceptron classi-
+fier with two hidden layers, 100 neurons each.
+4.4
+Hyperparameter tuning
+In preliminary experiments, we found that the most impor-
+tant hyperparameter is the threshold used for the informative-
+ness measure. For this reason, we focused on tuning this pa-
+rameter. We use random search [Bergstra and Bengio, 2012]
+to select the best thresholds for each algorithm. MLP clas-
+sifier was trained with Adam optimizer (learning rate equal
+to 0.001). We allow training for a maximum of 5000 itera-
+tions. Detailed description of hyperparameter tuning process
+with range of values for each algorithm are provided in ap-
+pendix D.
+4.5
+Goal of experiments
+The overall goal of experiments is to perform a thorough in-
+vestigation into the usefulness of self-labeling in a selective
+sampling setting. To provide a more precise description, we
+formulate the following research questions:
+RQ1: Is there a benefit of combining active learning strategies
+with self-labeling?
+RQ2: What is the performance of the proposed method for
+datasets with a high number of learning examples?
+RQ3: What is the impact of the initial training size (the seed
+size) on the performance?
+RQ4: How does the accuracy of the model trained with seed
+impact the learning process of the proposed algorithm?
+RQ5: Does the proposed algorithm allows for the better uti-
+lization of the computational budget?
+Each of these research questions will be addressed in the
+following parts of our work.
+5
+Experiments
+In this section, we describe the results of an experimental
+evaluation in accordance with the research questions stated
+above.
+5.1
+Experiments with smaller datasets
+We compare the performance of the proposed method and
+baselines according to the experimental protocol described in
+previous sections. Here we utilize four datasets, namely nurs-
+ery, mushroom, wine, and abalone. Results are presented on
+the left-hand side of Tab. 2.
+Here we can see that our method rarely obtains the best
+score. However, the difference between the best-performing
+method and SL2S is often close to or below 0.02. The worst
+performance is obtained for the nursery dataset. This is prob-
+ably due to the presence of three majority classes with a close
+number of samples and a single minority class with a sub-
+stantially lower number of samples. For this reason, a lot of
+samples could be discarded by the prior filter. Other datasets
+are either well-balanced or contain a single majority class.
+For these types of datasets, we obtained better results.
+When we compare the performance of other methods, we
+can notice that uncertainty-based methods perform compa-
+rably to the best algorithms only with a high budget. Clas-
+sification margin is a strong baseline as indicated by litera-
+ture [Bahri et al., 2022]. When we compare ensemble-based
+methods, it turns out that min margin and consensus entropy
+are the best, with both methods obtaining close performance
+to the best algorithm.
+5.2
+Experiments with bigger datasets
+We also conduct experiments on larger datasets, i.e., adult,
+bank marketing, firewall, and chess. To save computation,
+we train only when batch of 100 labeled samples is col-
+lected and reuse the hyperparameters found for the biggest
+datasets in previous experiments. We also drop the vote en-
+tropy and max disagreement methods from our comparison
+due to poor performance in previous experiments compared
+to other ensemble-based methods. Results are presented on
+the right-hand side of Tab. 2.
+Here our method performs well, with either the best-
+balanced accuracy or close to the best. There is no clear per-
+formance pattern when we compare results across the varying
+budget. Uncertainty-based methods provide the worst bal-
+anced accuracy in this case. Random sampling allows for
+obtaining the best performance for the firewall dataset, prob-
+ably due to the simplicity of the classification problem in this
+dataset. Firewall has only three classes and a lower IR com-
+pared to other datasets. Most methods perform well on this
+dataset, with a lot of ties between different algorithms in the
+first place.
+5.3
+Impact of seed size
+We evaluate the impact of the size of the initial training set
+on active learning performance. When utilizing labels gen-
+erated with model predictions, the lower number of initial
+training samples may cause a higher error rate at the begin-
+ning of the experiment and the introduction of more noise into
+the dataset. For this reason, smaller seed sizes can impact
+the overall results. We reuse the hyperparameter values from
+previous experiments. All experiments are performed with a
+budget equal to 0.3. The results are provided in appendix E.
+As expected, initial training size has a lower impact on the
+random sampling algorithm. This method is not dependent on
+model predictions, therefore, changing the seed should not
+impact the overall performance. In the case of uncertainty-
+based methods, there is no clear pattern of seed size impact.
+In some cases, training with these algorithms and lower seed
+could provide better results. The ensemble-based methods
+improve balanced accuracy as the number of labeled samples
+grows. SL2S can in some cases, obtain better performance
+with smaller seeds, and often we were able to obtain the best-
+balanced accuracy with our method. This result indicates that
+SL2S does not depend heavily on the initial model perfor-
+mance and could be applied even if the number of labeled
+samples in the beginning is small.
+5.4
+Ablation studies
+We perform ablation studies for the proposed method. First,
+we remove the prior filter and allow training regardless of
+
+Table 2: Balanced accuracy for variable budget size and smaller datasets
+dataset
+nursery
+adult
+labeled
+0.318±0.030
+0.741±0.010
+labeled ensemble
+0.276±0.013
+0.756±0.005
+budget
+0.1
+0.2
+0.3
+0.4
+0.5
+0.1
+0.2
+0.3
+0.4
+0.5
+random
+0.371±0.015
+0.350±0.017
+0.325±0.012
+0.298±0.017
+0.282±0.012
+0.735±0.007
+0.733±0.005
+0.729±0.007
+0.731±0.005
+0.732±0.006
+f. uncertainty
+0.389±0.018
+0.393±0.016
+0.385±0.007
+0.391±0.016
+0.394±0.019
+0.754±0.007
+0.758±0.008
+0.765±0.009
+0.760±0.011
+0.760±0.011
+v. uncertainty
+0.378±0.012
+0.359±0.012
+0.327±0.014
+0.307±0.014
+0.280±0.018
+0.756±0.013
+0.751±0.011
+0.755±0.012
+0.758±0.012
+0.746±0.011
+class. margin
+0.397±0.019
+0.395±0.020
+0.396±0.013
+0.399±0.036
+0.396±0.019
+0.757±0.008
+0.757±0.008
+0.757±0.008
+0.757±0.008
+0.757±0.008
+vote entropy
+0.393±0.014
+0.393±0.014
+0.393±0.014
+0.393±0.014
+0.393±0.014
+-
+-
+-
+-
+-
+consensus entropy
+0.393±0.013
+0.394±0.013
+0.393±0.014
+0.393±0.013
+0.404±0.017
+0.764±0.004
+0.767±0.005
+0.765±0.002
+0.765±0.004
+0.764±0.003
+max disagreement
+0.402±0.019
+0.393±0.014
+0.393±0.014
+0.404±0.016
+0.403±0.021
+-
+-
+-
+-
+-
+min margin
+0.405±0.019
+0.375±0.012
+0.388±0.021
+0.385±0.018
+0.400±0.010
+0.768±0.004
+0.767±0.005
+0.768±0.004
+0.768±0.004
+0.768±0.004
+SL2S
+0.384±0.018
+0.350±0.014
+0.338±0.013
+0.292±0.020
+0.294±0.016
+0.762±0.003
+0.763±0.004
+0.763±0.004
+0.762±0.003
+0.762±0.004
+dataset
+mushroom
+bank marketing
+labeled
+0.637±0.010
+0.712±0.012
+labeled ensemble
+0.636±0.010
+0.714±0.009
+budget
+0.1
+0.2
+0.3
+0.4
+0.5
+0.1
+0.2
+0.3
+0.4
+0.5
+random
+0.632±0.011
+0.634±0.009
+0.633±0.010
+0.636±0.010
+0.633±0.012
+0.700±0.023
+0.694±0.017
+0.703±0.021
+0.698±0.014
+0.699±0.018
+f. uncertainty
+0.630±0.010
+0.630±0.011
+0.633±0.012
+0.635±0.009
+0.634±0.010
+0.691±0.014
+0.710±0.014
+0.705±0.019
+0.705±0.019
+0.705±0.019
+v. uncertainty
+0.631±0.010
+0.631±0.012
+0.634±0.010
+0.633±0.009
+0.633±0.011
+0.690±0.014
+0.694±0.018
+0.700±0.018
+0.703±0.013
+0.697±0.018
+class. margin
+0.632±0.012
+0.633±0.012
+0.633±0.013
+0.618±0.023
+0.635±0.010
+0.682±0.012
+0.682±0.012
+0.682±0.012
+0.682±0.012
+0.682±0.012
+vote entropy
+0.630±0.011
+0.630±0.011
+0.635±0.011
+0.630±0.011
+0.635±0.010
+-
+-
+-
+-
+-
+consensus entropy
+0.632±0.010
+0.632±0.012
+0.633±0.011
+0.631±0.011
+0.634±0.010
+0.701±0.011
+0.714±0.007
+0.719±0.006
+0.715±0.009
+0.715±0.009
+max disagreement
+0.631±0.010
+0.633±0.013
+0.630±0.011
+0.630±0.011
+0.630±0.011
+-
+-
+-
+-
+-
+min margin
+0.632±0.011
+0.632±0.012
+0.634±0.010
+0.634±0.011
+0.634±0.011
+0.705±0.006
+0.716±0.007
+0.716±0.007
+0.716±0.007
+0.716±0.007
+SL2S
+0.631±0.011
+0.632±0.012
+0.632±0.012
+0.633±0.010
+0.634±0.010
+0.709±0.007
+0.715±0.009
+0.718±0.009
+0.719±0.008
+0.718±0.009
+dataset
+wine
+firewall
+labeled
+0.524±0.027
+0.997±0.001
+labeled ensemble
+0.514±0.015
+0.998±0.000
+budget
+0.1
+0.2
+0.3
+0.4
+0.5
+0.1
+0.2
+0.3
+0.4
+0.5
+random
+0.408±0.021
+0.430±0.021
+0.439±0.023
+0.452±0.018
+0.474±0.017
+0.996±0.002
+0.997±0.002
+0.998±0.001
+0.997±0.001
+0.998±0.001
+f. uncertainty
+0.418±0.020
+0.423±0.018
+0.441±0.017
+0.448±0.012
+0.440±0.012
+0.993±0.002
+0.996±0.002
+0.996±0.002
+0.996±0.002
+0.996±0.002
+v. uncertainty
+0.415±0.022
+0.437±0.016
+0.437±0.022
+0.459±0.021
+0.473±0.022
+0.994±0.002
+0.997±0.001
+0.997±0.001
+0.997±0.001
+0.997±0.001
+class. margin
+0.414±0.015
+0.433±0.016
+0.420±0.022
+0.424±0.014
+0.429±0.021
+0.991±0.001
+0.991±0.001
+0.991±0.001
+0.991±0.001
+0.991±0.001
+vote entropy
+0.404±0.012
+0.419±0.016
+0.389±0.016
+0.389±0.016
+0.389±0.016
+-
+-
+-
+-
+-
+consensus entropy
+0.419±0.010
+0.439±0.014
+0.458±0.014
+0.474±0.012
+0.479±0.013
+0.992±0.001
+0.992±0.001
+0.992±0.001
+0.992±0.001
+0.992±0.001
+max disagreement
+0.396±0.022
+0.389±0.016
+0.389±0.016
+0.389±0.016
+0.389±0.016
+-
+-
+-
+-
+-
+min margin
+0.420±0.014
+0.432±0.017
+0.461±0.019
+0.465±0.016
+0.487±0.012
+0.991±0.001
+0.991±0.001
+0.991±0.001
+0.991±0.001
+0.991±0.001
+SL2S
+0.420±0.014
+0.438±0.009
+0.451±0.022
+0.476±0.014
+0.499±0.019
+0.997±0.001
+0.997±0.001
+0.997±0.001
+0.997±0.001
+0.997±0.001
+dataset
+abalone
+chess
+labeled
+0.186±0.021
+0.816±0.011
+labeled ensemble
+0.188±0.012
+0.850±0.007
+budget
+0.1
+0.2
+0.3
+0.4
+0.5
+0.1
+0.2
+0.3
+0.4
+0.5
+random
+0.179±0.012
+0.177±0.009
+0.178±0.007
+0.185±0.011
+0.182±0.018
+0.515±0.017
+0.581±0.015
+0.630±0.009
+0.669±0.014
+0.698±0.009
+f. uncertainty
+0.171±0.024
+0.187±0.008
+0.177±0.013
+0.182±0.014
+0.186±0.018
+0.459±0.017
+0.538±0.015
+0.593±0.017
+0.639±0.017
+0.674±0.014
+v. uncertainty
+0.171±0.012
+0.182±0.014
+0.181±0.010
+0.176±0.006
+0.181±0.015
+0.464±0.021
+0.543±0.016
+0.604±0.016
+0.651±0.016
+0.691±0.014
+class. margin
+0.176±0.016
+0.181±0.008
+0.187±0.013
+0.177±0.018
+0.183±0.010
+0.466±0.019
+0.543±0.015
+0.606±0.013
+0.653±0.014
+0.692±0.014
+vote entropy
+0.185±0.015
+0.187±0.012
+0.185±0.015
+0.185±0.015
+0.188±0.012
+-
+-
+-
+-
+-
+consensus entropy
+0.188±0.015
+0.191±0.010
+0.185±0.013
+0.184±0.017
+0.187±0.014
+0.492±0.016
+0.594±0.009
+0.662±0.008
+0.715±0.010
+0.758±0.009
+max disagreement
+0.183±0.013
+0.185±0.012
+0.184±0.012
+0.185±0.015
+0.185±0.011
+-
+-
+-
+-
+-
+min margin
+0.184±0.009
+0.189±0.012
+0.185±0.015
+0.188±0.014
+0.184±0.014
+0.491±0.022
+0.589±0.017
+0.660±0.016
+0.713±0.016
+0.752±0.014
+SL2S
+0.183±0.013
+0.186±0.010
+0.190±0.010
+0.188±0.008
+0.182±0.011
+0.491±0.013
+0.585±0.010
+0.655±0.012
+0.702±0.010
+0.748±0.010
+the current dataset imbalance. This modification should fur-
+ther verify whether the dynamic imbalance is an issue when
+we use self-labeling in selective sampling. Secondly, we keep
+higher lambda values in equation 2 after the end of the budget.
+Decreasing lambda is the second mechanism introduced in
+our work that, in principle, should prevent the gradual degra-
+dation of model performance when using self-supervision as
+a source of new labels. Next, we remove the bootstrapped
+training to evaluate if ensemble diversification provides better
+performance in our experiments. Lastly, the self-labeling part
+of our approach was removed, and training was conducted
+with active learning alone. We use the wine dataset for eval-
+uation. Experiments were performed with a 0.3 labeling bud-
+get and various seed sizes. The prediction threshold value was
+selected based on hyperparameter tuning results from previ-
+ous experiments. We repeat experiments with three different
+random seeds and report average results in Tab. 3.
+Table 3: Balanced accuracy obtained in ablation study
+seed size
+100
+500
+1000
+base
+0.4151±0.0220
+0.4323±0.0230
+0.4509±0.0131
+-prior filter
+0.3747±0.0192
+0.4430±0.0193
+0.4502±0.0231
+-lambda reduction
+0.3747±0.0192
+0.4257±0.0159
+0.4463±0.0226
+-self-labeling
+0.4174±0.0168
+0.4421±0.0292
+0.4461±0.0398
+-bootstrapped training
+0.3916±0.0211
+0.4318±0.0217
+0.4461±0.0135
+We find that the prior filter has only a positive impact
+only in the case of smaller seed sizes. Conversely, reduc-
+ing lambda after budget end provides gains in balanced ac-
+curacy for higher seed size. Removing self-labeling increase
+accuracy. This finding is expected, as in preliminary experi-
+ments we found that na¨ıve application of self-labeling could
+make results worse. In this case, after removing two mecha-
+nisms from our algorithm that prevent the negative impact of
+dynamic imbalance and classification errors, we can see that
+
+Figure 3: Balanced accuracy (left) with a corresponding fraction of
+incorrect samples in the labeled dataset (right) over multiple itera-
+tions. We perform experiments for various seed sizes: 100 (top),
+500 (middle), and 1000 (bottom).
+further removing self-labeling improves the results. This is
+in line with preliminary results and further proves that prior
+filter and lambda reduction are indeed necessary. Lastly, the
+removal of bootstrapped training has a bigger impact when
+training with a smaller seed size.
+We can intuitively ex-
+plain this result by the fact that ensemble diversity should
+be smaller when utilizing bigger datasets, as more samples
+could cover feature space more densely, and randomly sam-
+pling datasets with bootstrapping would produce more similar
+datasets.
+5.5
+Incorrect labels from self-labeling
+As an extension of ablation studies, we examine how many
+wrong labels are introduced when using SL2S with and with-
+out prior filter. For this purpose, we train the MLP model on
+a wine dataset with a budget of 0.3 and various seed sizes.
+We plot the balanced accuracy with the corresponding frac-
+tion of samples with wrong labels in the training dataset over
+multiple iterations in Fig. 3. Including a prior filter drasti-
+cally reduces the number of incorrect labels. This does not
+necessarily lead to improvement in balanced accuracy. For a
+seed size equal to 500, both versions of the algorithm obtain
+very close final accuracy, while the difference in a fraction of
+incorrect labels is nearly 0.2. This phenomenon can be ex-
+plained by two factors. First is the fact that neural networks
+trained with gradient descent are known to be robust to noisy
+labels [Li et al., 2020]. Another possible explanation is that
+in some cases, incorrect labels could help to ”smooth” the de-
+cision boundary. This can also explain why no difference in
+balanced accuracy was observed only in part of our experi-
+ments. Nonetheless, more research is needed to better under-
+stand this phenomenon and its impact on self-labeling perfor-
+mance. Without prior filter, models trained with smaller seed
+sizes accumulate erroneous labels faster in the initial phase of
+training. For full SL2S the fraction of wrong labels roughly
+states the same across the whole training. We verify this fur-
+ther in appendix F and show that, indeed, initial model per-
+formance has no great impact on final accuracy of SL2S.
+6
+Lessons Learned
+Based on the results provided in Tab. 2 we can claim that
+SL2S method works better for big datasets. This was ex-
+pected, as for a larger stream, more samples can be accumu-
+lated with self-labeling. In the case of a smaller dataset, the
+performance is similar to other methods. The budget does
+not have a huge impact on the experiment results. Also, ex-
+periments with seed size confirm that our method could be
+applied for low data regimes.
+As indicated by results with the nursery dataset and abla-
+tion results, a prior filter may not be the best method to ad-
+dress the imbalance issue in our datasets. This part of the al-
+gorithm was designed with synthetic data. In the case of real
+datasets, the prior class distribution has higher importance.
+For this reason, alternative methods should be developed for
+dealing with imbalance when applying self-labeling to active
+learning scenarios. Although we did not manage to mitigate
+the imbalance problem properly, solving this issue is impor-
+tant, and future work in this area should address this problem.
+Results from Fig. 4 suggest that after the budget ends,
+the balanced accuracy roughly stays at the same level, and
+changes in the test accuracy do not occur frequently. With a
+higher number of model updates, the performance over time
+could fall drastically. For this reason, we introduced solutions
+that limit the use of self-labeling, preventing a fall in accu-
+racy. However, this is sub-optimal, as in this case, we would
+ideally want accuracy to increase over time, despite the end
+of the budget. More work is needed to better address the dy-
+namic imbalance issue or to provide a more accurate filter
+for wrong predictions. With these two problems solved we
+could give up the mechanisms that inhibit learning. It should
+allow for obtaining better performance, especially for bigger
+datasets.
+7
+Conclusions
+We have proposed a new active learning method that com-
+bines simple ensemble-based sample selection and self-
+labeling for selective sampling. Experiments with multiple
+baselines show that our method offers comparable perfor-
+mance to other active learning algorithms for smaller datasets
+and better performance for bigger datasets. Further experi-
+ments also show that our method could work well when the
+initially labeled dataset is small or when initial model accu-
+racy is poorly trained.
+We also show that an important aspect of self-labeling is
+an imbalance, as bias towards a single class in model predic-
+tions could, over time, increase dataset imbalance. Another
+important factor is erroneous model predictions that introduce
+noise into the training dataset. Based on the preliminaries and
+ablations presented in this work, we claim that further work
+should focus on these two aspects to improve the overall self-
+labeling performance. We cannot eliminate all errors from
+model predictions, however, developing better methods for
+filtering noisy labels or models that are more robust to label
+noise should allow for better utilization of self-labeling.
+
+0.40
+0.40
+SL2S
+of incorrect labels
+0.38
+nopriorfilter
+0.30
+0.36
+SL2S
+lanced
+0.34
+0.20
+no prior filter
+0.32
+bal
+0.10
+0.30
+0.28
+0.00
+0
+500
+1000
+1500
+2000
+25003000
+3500
+0
+500
+10001500
+2000
+25003000
+3500
+0.46
+0.20
+label
+accuracy
+0.44
+ofincorrect
+0.15
+0.42
+SL2S
+0.10
+0.40
+nopriorfilter
+0.38
+fraction
+0.05
+SL2S
+0.36
+nopriorfilter
+0.00
+0
+500
+1000
+1500
+2000
+2500
+3000
+0
+500
+1000
+1500
+2000
+2500
+3000
+SL2S
+of incorrect labels
+0.50
+SL2S
+0.13
+2
+no priorfilter
+no prior filter
+accura
+0.48
+0.10
+0.46
+0.08
+lanced
+0.44
+0.05
+fraction
+bal
+0.42
+0.03
+0.40
+0.00
+0
+500
+1000
+1500
+2000
+2500
+0
+500
+1000
+1500
+2000
+2500
+iterations
+iterationsAcknowledgment
+This work is supported by the CEUS-UNISONO programme,
+which has received funding from the National Science Centre,
+Poland under grant agreement No. 2020/02/Y/ST6/00037.
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+A
+Algorithm for selective sampling
+In this section, we provide a general algorithm for selective
+sampling, that depends on some informativeness measure m,
+that is determined by a specific active learning method. This
+algorithm can be easily expanded to include batch training.
+In such case, we introduce a buffer for labeled samples and
+train only where this buffer is full. After training, we empty
+the buffer.
+Algorithm 2 Selective sampling
+Require: L - set of labeled data, U - stream of N unlabeled
+samples, fθ - model, B - budget, m(.) - informativeness
+measure, α - threshold for informativeness measure
+1: train fθ on L
+2: for i ∈ {0, N} do
+3:
+ˆy ← fθ(xi)
+4:
+if B > 0 ∧ m(ˆy) ≤ α then
+5:
+request label y for xi
+6:
+L ← L ∪ {(xi, y)}
+7:
+train fθ on L
+8:
+B ← B − 1
+9:
+end if
+10: end for
+B
+Self-labeling Selective Sampling algorithm
+In this section, we provide the full SL2S algorithm in list-
+ing 3. First, we perform initial ensemble training with boot-
+strapping (lines 1-4). During sampling (line 2), we ensure
+that each sampled dataset has at least a single learning exam-
+ple from each class. Without this procedure, some software
+implementations of common classifiers could fail. Next, we
+sample unlabeled data from stream U. First, we gather predic-
+tions from the models in the ensemble (lines 6-8). We check
+if at least half of the models in the committee returned confi-
+dent and consistent predictions (lines (9-10). If the condition
+is met, we estimate the difference between the current estima-
+tion of the predicted class prior and prior for a perfectly bal-
+anced dataset (lines 11-13). If the current class prior does not
+exceed the 1
+C , we expand our dataset with the current sample
+labeled by prediction (line 14), calculate new λ value (lines
+15-20), and perform bootstrapped training (line 21) accord-
+ing to Algorithm 1. When the prediction is not consistent nor
+confident, we check if we still have the budget (line 24). If
+we do, a new query is created (line 25), and the dataset is up-
+dated with a new sample (line 26) and lambdas vector with a
+default value for labels obtained from oracle (line 27). Lastly,
+we perform bootstrapped training as in the previous case.
+Bootstrapped
+training
+average
+time
+complexity
+is
+O(L max(ME[r], Train(f)),
+where
+E[r]
+depends
+on
+average model supports, the value of τ, and utilization
+of budget, therefore for simplicity we left at as expected
+value.
+Self-labeling Selective Sampling time complexity
+is
+O(N max(LPred(f), LME[r], LTrain(f)),
+where
+Pred(f) and Train(f) are time complexities of model f
+prediction and training respectively.
+Algorithm 3 Self-labeling Selective Sampling
+Require: L - set of labeled data, U - stream of unlabeled
+data, {fθ}L - ensemble of L models, B - budget, τ - con-
+fident prediction threshold, k - number of newest samples
+from dataset used to estimate prior
+1: for l ∈ [0, L] do
+2:
+Sample dataset Dl by bootstrapping L
+3:
+train fθl on Dl
+4: end for
+5: for x ∈ U do
+6:
+for l ∈ [0, L] do
+7:
+ˆyl ← fθl(x)
+8:
+end for
+9:
+ˆy ← maxpl(y|x) ˆyl
+10:
+if �
+l 1maxc pl(yc|x)>τ > L
+2 ∧ ∀maxc pl(yc|x)>τ ˆyl =
+ˆy then
+11:
+ˆp ← 1
+k
+�M
+i=M−k 1yi=ˆy
+12:
+∆p = ˆp − 1
+C
+13:
+if ∆p ≤ 0 then
+14:
+L ← L ∪ {(x, ˆy)}
+15:
+λ = maxl,c pl(yc|x)
+τ
+− 1B=0
+16:
+λ ← (λ, λ)
+17:
+Bootstrapped training(L, {fθ}L, λ)
+18:
+end if
+19:
+else
+20:
+if B > 0 then
+21:
+request label y for x
+22:
+L ← L ∪ {(x, y)}
+23:
+λ ← (λ, 1)
+24:
+B ← B − 1
+25:
+Bootstrapped training(L, {fθ}L, λ)
+26:
+end if
+27:
+end if
+28: end for
+
+C
+Data lodaing procedures
+During data preprocessing, we split features into numerical
+and categorical features. For numeric features, we replace
+missing values with the median of a given feature and per-
+form standard scaling by subtracting the mean and scaling
+to unit variance. For categorical features, we utilize only a
+one-hot encoder. Detailed features and their categorization is
+provided in Tab. 5.
+Some classes have a too low number of samples com-
+pared to other classes to allow for learning. For example,
+in the case when five classes have above 1000 learning ex-
+amples each, and one class has only five samples, there is no
+point in keeping this class in the dataset. For this reason, we
+drop classes with the lowest number of samples. The list of
+dropped classes for each dataset is presented in Tab. 4.
+Table 4: Classes dropped from datasets
+dataset
+dropped classes
+adult
+-
+bank marketing
+-
+firewall
+reset-both
+chess
+zero, one, three
+nursery
+recommend
+poker
+7, 8, 9
+mushroom
+-
+wine
+3, 9
+abalone
+32, 20, 3, 21, 23, 22,
+27, 24, 1, 26, 29, 2, 25
+The classification scores were too high for the nursery and
+mushroom datasets. For this reason, experimental evaluation
+of active learning algorithms is nearly impossible, as each al-
+gorithm could easily obtain a perfect or near-perfect score.
+To prevent this, we make the classification task harder by
+dropping the most informative features, selected based on the
+Person correlation coefficient computed between features and
+labels. The complete list of features used and target columns
+used in experiments are provided in Tab. 5 and 6 for small
+and big datasets respectively.
+Table 5: Detailed categorization of features and classes dropped
+from small datasets
+dataset
+numeric features
+categorical features
+target feature
+adult
+age, education-num,
+capital-gain, fnlwgt,
+capital-loss, hours-
+per-week
+workclass,
+educa-
+tion, marital-status,
+occupation,
+rela-
+tionship, race, sex,
+native-country
+earnings
+bank
+marketing
+age, duration, cam-
+paign,
+pdays, previous
+job, marital, educa-
+tion, default,
+housing, loan, con-
+tact, month, pout-
+come
+y
+firewall
+Source Port,
+Des-
+tination Port, pkts
+sent,
+NAT Source
+Port,
+NAT
+Desti-
+nation Port, Bytes,
+Bytes Sent,
+Pack-
+ets, Bytes Received,
+Elapsed Time (sec),
+pkts received
+-
+Action
+chess
+0, 1, 2, 3, 4, 5
+-
+6
+Table 6: Detailed categorization of features and classes dropped
+from big datasets
+dataset
+numeric features
+categorical features
+target feature
+adult
+age, education-num,
+capital-gain, fnlwgt,
+capital-loss, hours-
+per-week
+workclass,
+educa-
+tion, marital-status,
+occupation,
+rela-
+tionship, race, sex,
+native-country
+earnings
+bank
+marketing
+age, duration, cam-
+paign,
+pdays, previous
+job, marital, educa-
+tion, default,
+housing, loan, con-
+tact, month, pout-
+come
+y
+firewall
+Source Port,
+Des-
+tination Port, pkts
+sent,
+NAT Source
+Port,
+NAT
+Desti-
+nation Port, Bytes,
+Bytes Sent,
+Pack-
+ets, Bytes Received,
+Elapsed Time (sec),
+pkts received
+-
+Action
+chess
+0, 1, 2, 3, 4, 5
+-
+6
+D
+Hyperparameter optimization process
+We perform hyperparameter tuning with random sampling
+[Bergstra and Bengio, 2012].
+Each method has the same
+search budget, i.e., the same number of runs. We sample
+20 values from the predefined prediction thresholds for each
+method. Interval borders are provided in Tab. 7. For each
+prediction threshold, we perform three evaluations with dif-
+ferent random seeds. We select the best hyperparameter val-
+ues based on averaged accuracy from these three runs. Ac-
+curacy reported in the work was obtained with ten random
+seeds, different than the ones used in the hyperparameter tun-
+ing process.
+Table 7: Intervals used for hyperparameters search process.
+method
+min value
+max values
+fixed uncertainty
+0.5
+1.0
+variable uncertainty
+0.5
+1.0
+classification margin
+0.0
+0.8
+vote entropy
+1.0
+50.0
+consensus entropy
+0.1
+1.0
+max disagreement
+1.0
+20.0
+min margin
+0.0
+0.5
+ours
+0.5
+1.0
+E
+Results of experiments with different seed
+size
+We provide full results of experiments with seed size Tab. 8.
+A description of these results is provided in the main part of
+our paper.
+F
+Impact of initial model accuracy
+In previous experiments, we showed that initial training set
+size does not have a large impact on SL2S performance.
+However, our primary concern are erroneous predictions pro-
+duced by a weak model. For this reason, we study the rela-
+tionship between the balanced accuracy of the initial model
+and overall experiment results. We gradually increase the
+seed size from 10 samples up and track test accuracy. When
+test accuracy exceeds the predefined value, we stop initial
+
+Table 8: Balanced accuracy for various seed sizes used for initial
+training of the model.
+dataset nursery
+all labeled
+0.318±0.030
+all labeled ensemble
+0.276±0.013
+seed size
+100
+500
+1000
+random
+0.341±0.017
+0.327±0.009
+0.325±0.012
+fixed uncertainty
+0.386±0.014
+0.388±0.012
+0.385±0.007
+variable uncertainty
+0.341±0.011
+0.338±0.016
+0.327±0.014
+classification margin
+0.382±0.012
+0.402±0.021
+0.396±0.013
+vote entropy
+0.354±0.010
+0.394±0.011
+0.393±0.014
+consensus entropy
+0.368±0.011
+0.389±0.012
+0.393±0.014
+max disagreement
+0.356±0.012
+0.401±0.010
+0.393±0.014
+min margin
+0.394±0.019
+0.358±0.012
+0.388±0.021
+SL2S
+0.361±0.009
+0.353±0.016
+0.338±0.013
+dataset mushroom
+all labeled
+0.637±0.010
+all labeled ensemble
+0.636±0.010
+seed size
+100
+500
+1000
+random
+0.633±0.007
+0.631±0.010
+0.633±0.010
+fixed uncertainty
+0.627±0.011
+0.629±0.010
+0.633±0.012
+variable uncertainty
+0.622±0.013
+0.631±0.011
+0.634±0.010
+classification margin
+0.620±0.016
+0.630±0.013
+0.633±0.013
+vote entropy
+0.632±0.012
+0.624±0.012
+0.635±0.011
+consensus entropy
+0.633±0.011
+0.629±0.013
+0.633±0.011
+max disagreement
+0.596±0.023
+0.624±0.012
+0.630±0.011
+min margin
+0.628±0.009
+0.631±0.010
+0.634±0.010
+SL2S
+0.635±0.009
+0.632±0.012
+0.632±0.012
+dataset wine
+all labeled
+0.524±0.027
+all labeled ensemble
+0.514±0.015
+seed size
+100
+500
+1000
+random
+0.403±0.021
+0.418±0.018
+0.439±0.023
+fixed uncertainty
+0.395±0.017
+0.421±0.020
+0.441±0.017
+variable uncertainty
+0.406±0.019
+0.423±0.019
+0.437±0.022
+classification margin
+0.355±0.011
+0.414±0.015
+0.420±0.022
+vote entropy
+0.291±0.014
+0.428±0.022
+0.389±0.016
+consensus entropy
+0.407±0.017
+0.427±0.013
+0.458±0.014
+max disagreement
+0.316±0.026
+0.356±0.020
+0.389±0.016
+min margin
+0.401±0.012
+0.435±0.022
+0.461±0.019
+SL2S
+0.405±0.011
+0.439±0.024
+0.451±0.022
+dataset abalone
+all labeled
+0.186±0.021
+all labeled ensemble
+0.188±0.012
+seed size
+100
+500
+1000
+random
+0.180±0.013
+0.180±0.011
+0.178±0.007
+fixed uncertainty
+0.175±0.015
+0.189±0.020
+0.177±0.013
+variable uncertainty
+0.182±0.013
+0.186±0.009
+0.181±0.010
+classification margin
+0.178±0.013
+0.180±0.014
+0.187±0.013
+vote entropy
+0.187±0.014
+0.189±0.011
+0.185±0.015
+consensus entropy
+0.184±0.009
+0.190±0.012
+0.185±0.013
+max disagreement
+0.166±0.017
+0.180±0.010
+0.184±0.012
+min margin
+0.179±0.018
+0.191±0.013
+0.185±0.015
+SL2S
+0.187±0.022
+0.184±0.013
+0.190±0.010
+training and proceed to the further part of selective sampling.
+We evaluate models with initial accuracy equal to 0.15, 0.2,
+0.25, 0.3, 0.35, and 0.4, and with a budget of 0.3. Results are
+plotted in Fig. 4.
+From this plot, we deduce that low accuracy at the begin-
+ning can be easily compensated by spending the budget on
+initial model improvement. Looking at the balanced accuracy
+values over time, we can see that initial values of accuracy for
+poorly trained models increase abruptly and the beginning of
+training. Higher initial budget consumption can be further ex-
+emplified by the iteration number when the budget runs out.
+For larger initial accuracy, the budget end is later. It shows
+that our method can be used even with a poorly trained initial
+model.
+Figure 4: Impact of initial balanced accuracy on overall training re-
+sults. Vertical doted lines indicate the iteration when budget ended.
+
+0.50
+0.45
+0.40
+accuracy
+0.35
+lanced
+0.30
+bal
+0.15
+0.25
+0.2
+0.25
+0.3
+0.20
+0.35
+0.4
+0
+500
+1000
+1500
+2000
+2500
+iterations

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A9FAT4oBgHgl3EQfrx7P/content/tmp_files/2301.08655v1.pdf.txt

@@ -0,0 +1,1366 @@
+arXiv:2301.08655v1  [math.OA]  20 Jan 2023
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM
+ANNULUS
+SLAWOMIR KLIMEK, MATT MCBRIDE, AND KAORU SAKAI
+Abstract. We construct compact parametrix implementations of covariant derivations on
+the quantum annulus.
+1. Introduction
+The goal of this paper is to provide simple examples of Dirac type operators on noncom-
+mutative compact manifolds. We study analogs of d-bar operators on the quantum annulus
+using only inherent geometrical structures: rotations, invariant states, covariant derivations
+and their implementations. In our previous paper [3], we constructed similar d-bar type
+operators on the quantum annulus using APS-type boundary conditions. The class of oper-
+ators was designed to mimic the classical Atiyah-Patodi-Singer theory and is different, less
+geometrical than the class studied in this paper. The main outcome, as in the past paper,
+is that we show that our quantum d-bar type operators have compact parametrices, like
+elliptic differential operators on compact manifolds.
+Our another paper [5] on quantum annulus, following [4], contains a description of un-
+bounded derivations, covariant with respect to a natural rotation, and their implementations
+in Hilbert spaces obtained from the GNS construction with respect to invariant states. It
+turned out that no such implementation in any GNS Hilbert space for a faithful, normal,
+invariant state has compact parametrices for a large class of boundary conditions. However,
+as demonstrated in [6], if we relax the concept of an implementation by allowing operators
+to act between different Hilbert spaces, then there is an interesting class of examples of
+quantum d-bar operators with compact parametrices that can be constructed this way for
+the case of the quantum disk. It is the purpose of this paper to extend those ideas to the
+quantum annulus.
+Spectral triples are a key tool in noncommutative geometry [1], as they allow using ana-
+lytical methods in studying quantum spaces. Since compact parametrix property is a part
+of the spectral triples conditions, our papers [4] and [5] demonstrate, using analytic tech-
+niques, that spectral triples, in general, cannot be constructed on the quantum disk and
+the quantum annulus using implementations of covariant derivations in GNS Hilbert spaces.
+Another, topological reason was pointed out in [7], in the case of the quantum disk T .
+Namely, the pull-back map in K-Homology K0(C(S1)) → K0(T ) is an isomorphism and so
+the restriction map K0(T ) → K0(K) is a zero map. Consequently, any spectral triple over
+the Toeplitz algebra, when restricted to the ideal of compact operators K should be trivial in
+K-Homology. However, it is easy to compute that implementations of covariant derivations
+Date: January 23, 2023.
+1
+
+2
+KLIMEK, MCBRIDE, AND SAKAI
+pair nontrivially with a minimal projection in K, and hence they cannot lead to spectral
+triples over T . Similar arguments seem to also apply to the quantum annulus.
+Additionally, as pointed out in [2], there are fundamental reasons why APS boundary
+conditions are not compatible with spectral triples even in classical geometry for algebras
+of functions which are non-constant on the boundary, as the corresponding domains of the
+Dirac-type operators are not preserved by the representations of the algebra.
+In [6], the authors claimed to construct an even spectral triple over the quantum disk. Due
+to technicalities in the definition of an implementation of a derivation, however, this was not
+true. We clarify the generalized concept of implementations of unbounded derivations and
+when they lead to spectral triples in the next section. In Section 3 we establish the notation
+and review results from [5]. The main result, Theorem 4.3, is proved in Section 4. It states
+that, for a class of exponential coefficients, the operator D defined in equation (3.6) as a
+suitable Hilbert spaces implementation of a covariant derivation δ of (3.2), in the quantum
+annulus algebra A, see (3.1), has a compact parametrix; in fact the inverse of D is compact.
+2. Implementations of Unbounded Derivations
+Let A be a C∗-algebra, A ⊆ A a dense ∗-subalgebra and δ : A �→ A a derivation. Suppose
+that H1 and H2 are Hilbert spaces carrying representations of A and denoted π1 and π2
+respectively.
+The following is a natural concept of an implementation of a derivation in
+A between two Hilbert spaces, generalizing the usual notion of an implementation of an
+unbounded derivation.
+An implementation of δ between H1 and H2 consists of the following:
+• A dense subspace dom(D) ⊆ H1
+• An implementing operator D : dom(D) → H2
+• An intertwinner i : dom(D) → H2
+such that
+(1) ∀a ∈ A, ∀x ∈ dom(D)
+π1(a)x ∈ dom(D)
+(2) ∀a ∈ A, ∀x ∈ dom(D)
+i(π1(a)x) = π2(a)i(x)
+(3) ∀a ∈ A, ∀x ∈ dom(D)
+Dπ1(a)x − π2(a)Dx = π2(δ(a))i(x)
+A special case of the above definition is when H1 = H2 = H, π1 = π2 = π and i is
+the identity map. Then the second condition above is obviously satisfied, while the third
+condition can be written as
+(Dπ(a) − π(a)D)x = π(δ(a))x.
+This coincides with the usual concept of an unbounded implementation of a derivation as a
+commutator.
+Recall that a closed operator D is called a Fredholm operator if there are bounded op-
+erators Q1 and Q2 such that Q1D − I and DQ2 − I are compact. The operators Q1 and
+Q2 are called left and right parametrices respectively. We say that a Fredholm operator D
+has compact parametrices if at least one (and consequently both) of the parametrices Q1
+
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS
+3
+and Q2 is compact. More on general properties of operators with compact parametrices can
+be found in the appendix of [4]. We also say that an implementation (dom(D), D, i) has
+compact parametrices if the closure of the operator D has compact parametrices.
+Under additional conditions an implementation of a derivation can lead to an even spectral
+triple over A. Namely, defining H = H1
+� H2, with grading Γ
+��
+H1 = 1 and Γ
+��
+H2 = −1 and a
+representation π : A → B(H) of A in H given by the formula:
+π(a) = (π1(a), π2(a)),
+and also defining a generally unbounded operator D in H by:
+D =
+�
+0
+D
+D∗
+0
+�
+,
+we see that π(a) are even and D is odd with respect to grading Γ. If D is a self-adjoint
+operator with compact parametrix and additionally the intertwinner i is bounded, then the
+conditions in the definition of a derivation implementation imply that π(a) preserve the
+domain of D for all a ∈ A and the commutator [D, π(a)] is bounded as can be seen by
+a simple calculation. Consequently, under those additional conditions, we obtain an even
+spectral triple over A.
+A very natural class of implementations of derivations can be obtained from GNS rep-
+resentations in the following way. Suppose τ1, τ2 are faithful states on A. Let H1, H2 be
+the corresponding GNS Hilbert spaces, obtained by completing A with respect to the inner
+products
+(a, b)i = τi(a∗b),
+i = 0, 1.
+Because we assume that the states are faithful, A sits densely in H1, H2. More precisely,
+there are injective continuous linear maps φ1 : A → H1, φ1 : A → H1 with dense ranges
+embedding A into H1, H2.
+The Hilbert spaces H1, H2 carry natural representations of A given by left multiplication;
+for a, b ∈ A we have
+πi(a)φi(b) = φi(ab).
+Suppose as before that we have a dense ∗-subalgebra A ⊆ A and a derivation δ : A �→ A.
+Then we have the following implementation of δ between H1 and H2:
+• dom(D) := φ1(A) ⊆ H1. If x ∈ dom(D) then we write x = φ1(b) for some b ∈ A,
+notation we use in the formulas below.
+• D(x) := φ2(δ(b)),
+• i(x) := φ2(b)
+It is a matter of straightforward calculations to verify that indeed the three conditions of the
+definition are satisfied and the above defines an implementation of δ between H1 and H2.
+3. Quantum Annulus Preliminaries
+We review the notation and basic concepts from [5] below and, in a number of places, we
+use the results contained in that paper.
+
+4
+KLIMEK, MCBRIDE, AND SAKAI
+3.1. The Quantum Annulus. Let {El}l∈Z be the canonical basis for ℓ2(Z) and V be the
+bilateral shift defined by
+V El = El+1 .
+Notice that V is a unitary. Let L be the diagonal label operator defined by
+LEl = lEl .
+It follows from the functional calculus that given a function a : Z → C, we have
+a(L)El = a(l)El .
+These are precisely the operators which are diagonal with respect to {El}. The operators
+(L, V ) serve as noncommutative polar coordinates, and they satisfy the following commuta-
+tion relation:
+LV = V (L + I) .
+Let c(Z) be the set of a(l), as above, which are convergent as l → ±∞ and let c++
+00 (Z) be the
+set of all eventually constant functions, i.e. functions a(l) such that there exists a l0 where
+a(l) is a constant for l ≥ l0 and also is a possibly different constant for l ≤ −l0.
+Let A be the C∗-algebra generated by V and a(L), that is:
+A = C∗(V, a(L) : a(l) ∈ c(Z))
+(3.1)
+This algebra is called the quantum annulus. The smallest reasonable domain of derivations
+in A is the following dense ∗-subalgebra of A:
+A =
+�
+a =
+�
+n∈Z
+V nan(L) : an(l) ∈ c++
+00 (Z), finite sums
+�
+.
+3.2. Derivations in the Quantum Annulus. Let ρθ : A → A, 0 ≤ θ < 2π, be a one
+parameter group of automorphisms of A defined by:
+ρθ(a) = eiθLae−iθL.
+Since ρθ(a(L)) = a(L), ρθ(V ) = eiθV and consequently ρθ(V −1) = e−iθV −1, the auto-
+morphisms ρθ are well defined on A and they preserve A. By Proposition 3.2 in [5], any
+densely-defined derivation δ : A → A, covariant with respect to ρθ that is
+ρθ(δ(a)) = eiθδ(ρθ(a)) ,
+is of the following form:
+δ(a) = [Uβ(L), a]
+(3.2)
+where {β(l + 1) − β(l)} ∈ c(Z). We use notation:
+lim
+l→±∞(β(l + 1) − β(l)) := β±∞,
+and below we only consider covariant derivations with β±∞ ̸= 0. It follows that there are
+constants c1 and c2 so that
+c1(|l| + 1) ≤ |β(l)| ≤ c2(|l| + 1) .
+(3.3)
+
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS
+5
+3.3. Covariant Implementations on the Quantum Annulus. Here we consider covari-
+ant implementations of derivations (3.2). We begin by introducing the following family of
+states τw : A → C on A, defined by
+τw(a) = tr(w(L)a) ,
+where w(l) > 0 for all l ∈ Z and
+�
+l∈Z
+w(l) = 1 .
+As a result of Proposition 4.3 in [5], τw are precisely the ρθ-invariant, normal, faithful states
+on A. Let Hw be the Hilbert space obtained by Gelfand-Naimark-Segal (GNS) construction
+on A using state τw. Since the state is faithful, Hw is the completion of A with respect to
+the inner product given by
+⟨a, b⟩w = τw(a∗b) .
+A simple calculation leads to the following precise description:
+Hw =
+�
+f =
+�
+n∈Z
+V nfn(L) : ∥f∥2
+w =
+�
+n∈Z
+�
+l∈Z
+w(l)|fn(l)|2 < ∞
+�
+.
+(3.4)
+With this identification we naturally have A ⊆ Hw so the inclusion maps φw : A → Hw
+are the identity maps. Notice also that A is dense in Hw. The GNS representation map
+πw : A → B(Hw) is given by left-hand multiplication:
+πw(a)f = af .
+Define a one parameter group of unitary operators V w
+θ : Hw → Hw via the formula:
+V w
+θ f =
+�
+n∈Z
+V neinθfn(L) .
+An immediate calculation shows that:
+πw(ρθ(a)) = V w
+θ πw(a)(V w
+θ )−1 ,
+and therefore the operators V w
+θ are implementing the one parameter group of automorphisms
+ρθ.
+Consider an additional weight, w′(l), possibly different from w(l), satisfying the same
+conditions. Proceeding like in the previous section, we set
+dom(D) := A ⊂ Hw ,
+and choose for an implementing operator
+i : dom(D) = A → A ⊂ Hw′
+to be the identity operator a �→ a. Clearly, the first two properties of an implementation are
+satisfied. We say that an operator D : Hw ⊇ A → Hw′ defines a covariant implementation
+of the covariant derivation (3.2) if for every a ∈ A, and for every f ∈ A considered as an
+element of both Hw and Hw′, we have:
+Dπw(a)f − πw′(a)Df = πw′(δ(a))f ,
+and, additionally, D satisfies:
+V w′
+θ D(V w
+θ )−1f = eiθDf .
+
+6
+KLIMEK, MCBRIDE, AND SAKAI
+Proceeding as in Proposition 5.2 in [5] shows the following result.
+Proposition 3.1. There exists a sequence {α(l)} satisfying
+�
+l∈Z
+|β(l) − α(l)|2w′(l) < ∞
+(3.5)
+such that any covariant implementation D : Hw ⊇ A → Hw′ is of the form:
+Df = V β(L)f − fV α(L) .
+(3.6)
+Conversely, for any {α(l)} satisfying (3.5), the formula (3.6) defines a covariant implemen-
+tation D : Hw ⊇ A → Hw′ of the derivation (3.2).
+We assume below that for every l we have:
+α(l), β(l) ̸= 0 .
+It is convenient, like in [4] and [5], to write
+α(l) = β(l)µ(l + 1)
+µ(l)
+for some sequence {µ(l)} such that µ(0) = 1.
+3.4. Fourier Decomposition. To further analyze the operator D of formula (3.6), we can
+decompose it into a Fourier series and study its Fourier components which are operators
+acting between weighted ℓ2-spaces, defined as follows:
+ℓ2
+w =
+�
+{f(l)}l∈Z :
+�
+l∈Z
+|f(l)|2w(l) < ∞
+�
+.
+We have the following decomposition proposition:
+Proposition 3.2. Let f ∈ dom(D). Then
+Df =
+�
+n∈Z
+V n+1(Dnfn)(L) ,
+where Dn : ℓ2
+w ⊇ c++
+00 (Z) → ℓ2
+w′ and is given by the following formula:
+(Dnh)(l) = β(l + n)h(l) − β(l)µ(l + 1)
+µ(l)
+h(l + 1)
+for some h ∈ c++
+00 (Z).
+Proof. The proof follows by writing f ∈ dom(D) as its Fourier series, applying D to it and
+using the commutation relation LV = V (L + I).
+□
+In what follows, the purpose is to choose the parameters such that D has a compact
+parametrix.
+
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS
+7
+3.5. Parametrices. Next we study a formal candidate for a parametrix of D. The closure of
+D, defined as above on c++
+00 (Z), will be denoted by ¯D while its closure defined on the space
+c00(Z) of eventually zero functions will be denoted by ¯D00. Also notice that D preserves
+c00(Z). We have the following simple observation.
+Proposition 3.3. If {α(l)} satisfies (3.5) then ¯D = ¯D00.
+Proof. Notice that c00(Z) ⊂ dom(D) is a co-dimension 2 subspace. Thus, it is enough to
+verify that 1 and the characteristic function of Z≥0 are in the domain of ¯D. Approximating
+1 by characteristic functions χN of sets −N ≤ l ≤ N, χN ∈ c00(Z), we see that D(χN)
+converges in ℓ2
+w to D(1), which is in ℓ2
+w by (3.5), implying that 1 is in the closure of D. The
+characteristic function of Z≥0 is in the domain of ¯D by the same argument.
+□
+Let Qn be given by the following formula:
+(Qng)(l) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+∞
+�
+j=l
+�l+n−1
+k=l
+β(k)
+�j+n
+k=j β(k)
+· µ(j)
+µ(l) g(j)
+n ≥ 0
+∞
+�
+j=l
+�j−1
+k=j+n+1 β(k)
+�l−1
+k=l+n β(k)
+· µ(j)
+µ(l) g(j)
+n < 0.
+This expression was obtained by inverting Dn using techniques similar to the calculations in
+Proposition 4.12 in [3]. Notice that we have:
+Qn : c00(Z) → dom(D),
+since the sums in the definitions of Qn are finite for g ∈ c00(Z) and the outcomes are
+eventually constant. Relations between Dn and Qn are explained in the following statements.
+Proposition 3.4. For every f ∈ c00(Z) we have:
+DnQnf = f
+and QnDnf = f .
+Proof. The formulas follow from straightforward calculations.
+□
+From this proposition we can formally define the inverse for D to be Q = �
+n∈Z Qn. For
+this to be well-defined, the series needs to converge. In fact, once we verify that Q is bounded,
+the previous two propositions imply that Q is the inverse of ¯D.
+4. Results
+For the remainder of this section we assume that β(k) = k + 1
+2. Moreover we only consider
+the special choices of the weights {w(l)}, {w′(l)} and the choice for {µ(l)} namely:
+w(l) = e−a|l|,
+w′(l) = e−b|l|,
+and µ(l) = e−(γl)/2, for a, b, γ > 0 .
+It should be noted that with these specific choices α(l) = e−γ/2β(l) and the conditions
+in Proposition 3.1 are trivially satisfied.
+By a simple perturbative argument the results
+below are valid for a much larger class of coefficients, see the remark at the end of the next
+subsection.
+
+8
+KLIMEK, MCBRIDE, AND SAKAI
+4.1. Compactness of Parametrices. We show that Qn are Hilbert-Schmidt operators for
+every n and verify that their respective Hilbert-Schmidt norms go to zero as n goes to infinity,
+implying that Q is a compact operator. To prove this we need a few helper lemmas. We
+postpone proofs of those lemmas until the next subsection.
+Lemma 4.1. For 0 ≤ l ≤ n, define the following product:
+qn(l) =
+� 1
+2
+�2 � 3
+2
+�2 · · ·
+�
+n − 1
+2
+�2
+�
+l − 1
+2
+�2 �
+l − 3
+2
+�2 · · ·
+�
+l − n + 1
+2
+�2 .
+Then we have the identity:
+qn(l) = ((2n − 2l + 1) · · ·(2n − 3)(2n − 1))2
+(1 · 3 · 5 · · ·(2l − 1))2
+for every natural number n .
+Moreover, qn(l) satisfies the following three estimates:
+(1) qn(l) ≥ 1,
+(2) qn(l) ≤ 2l
+�
+2n
+2l
+�
+,
+(3) qn(j)
+qn(l) ≤
+�
+2n
+2l
+�
+for 0 ≤ j ≤ l .
+Lemma 4.2. For a nonnegative integer j, define the following sum:
+Jn(j) =
+�
+k≥0
+�
+k + j + 1
+2
+�2 · · ·
+�
+k + j + n − 1
+2
+�2 e−γk
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2
+.
+Then for n ≥ 0
+Jn(j) ≤
+2n + 1
+�
+1 − e− γ
+2 �2n+1 .
+The following is the main technical result of the paper.
+Theorem 4.3. Suppose that γ > a > b and that
+exp
+�
+−(a − b)
+2
+�
++ exp
+�
+−γ + a
+2
+�
+< 1 .
+Then Qn : ℓ2
+w′ → ℓ2
+w is a Hilbert-Schmidt operator for every n ∈ Z and ∥Qn∥HS → 0 as
+n → ±∞. Consequently, Q = �
+n Qn is the inverse of ¯D and is a compact operator.
+Proof. Notice that formula for Qn shows that it is an integral operator. Therefore, by direct
+calculation we can compute the Hilbert-Schmidt norms:
+∥Qn∥2
+HS =
+
+
+
+
+
+
+
+
+
+
+
+
+
+∞
+�
+j=l
+�l+n−1
+k=l
+|β(k)|2
+�j+n
+k=j |β(k)|2 · |µ(j)|2
+|µ(l)|2 · w(l)
+w′(j)
+n ≥ 0
+∞
+�
+j=l
+�j−1
+k=j+n+1 |β(k)|2
+�l−1
+k=l+n |β(k)|2
+· |µ(j)|2
+|µ(l)|2 · w(l)
+w′(j)
+n < 0
+Since the choice of β’s are linear, the following ratios:
+����
+β(l) · · ·β(l + n − 1)
+β(l + n) · · ·β(l − 1)
+����
+2
+and
+����
+β(j + n + 1) · · ·β(j − 1)
+β(l + n) · · ·β(l − 1)
+����
+2
+
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS
+9
+are ratios of polynomials. Thus by our choice of exponential µ’s, w’s and w′’s, ∥Qn∥HS exists
+if and only if
+�
+j≥l
+����
+µ(j)
+µ(l)
+����
+2
+· w(l)
+w′(j) < ∞ .
+However, this easily follows if γ > a > b. Thus the Hilbert-Schmidt norm of Qn exists for
+all n ∈ Z. It remains to show that those norms go to zero as n → ±∞. We only need to
+study the case for n ≥ 0 as if n < 0 then by doing a change of variables of j �→ −l, l �→ −j
+and n �→ −n − 1 we are back in the n ≥ 0 case. Thus, we only need to estimate the sum:
+∥Qn∥2
+HS =
+�
+j≥l
+�
+l + 1
+2
+�2 · · ·
+�
+l + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2 · e−γ(j−l)−a|l|+b|j|
+�
+j + n + 1
+2
+�2 .
+The sum is over all j ≥ l. It splits into sums over four main regions which we will further
+subdivide as illustrated in the picture below:
+We have
+∥Qn∥2
+HS ≤ SA + SB + SC + SD,
+as for simplicity of estimations we let the regions overlap. Here the regions are: region A:
+j ≥ l ≥ 0, region B: j ≥ −l ≥ 0, l ≤ 0, region C: −l ≥ j ≥ 0, l ≤ 0, and region D: l ≥ j ≥ 0,
+which we will handle separately. In our estimates, we double count the boundaries in some
+places for convenience of different estimates.
+Regions A and B: Notice the following observation: if −j ≤ l ≤ j for j ≥ 0, then for
+any r > 0 we have that (l + r)2 ≤ (j + r)2. Using this fact, we can estimate:
+�
+l + 1
+2
+�2 · · ·
+�
+l + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2 ≤ 1.
+
+B
+A
+C3
+C2
+C1
+D3
+D1
+D210
+KLIMEK, MCBRIDE, AND SAKAI
+It follows that we have
+SA ≤
+�
+j≥l≥0
+e−γ(j−l)−al+bj
+�
+j + n + 1
+2
+�2 → 0 as n → ∞ .
+Changing l → −l we get exactly the same estimate for SB, and so SB → 0 as n → ∞.
+Region C: 0 ≤ j ≤ −l.
+First we map l �→ −l, then we split this region into two sub-regions: C1: 0 ≤ j ≤ l ≤ n
+and the complement of C1: 0 ≤ j ≤ l, l ≥ n. In the second region we make a change of
+variables of l′ = l − n, then replace l′ with l and we further get two additional sub-regions:
+C2: 0 ≤ j ≤ l and C3: 0 ≤ l ≤ j ≤ l + n.
+First in sub-region C1: since l ≤ n, using the first inequality from Lemma 4.1, we have
+SC1 =
+�
+0≤j≤l≤n
+�
+l − 1
+2
+�2 · · ·
+�
+l − n + 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2 · e−γ(j+l)−al+bj
+�
+j + n + 1
+2
+�2
+≤
+�
+0≤j≤l≤n
+�
+l − 1
+2
+�2 · · ·
+�
+l − n + 1
+2
+�2
+�1
+2
+�2 · · ·
+�
+n − 1
+2
+�2
+· e−γ(j+l)−al+bj
+�
+j + n + 1
+2
+�2
+≤
+�
+0≤j≤l≤n
+e−γ(j+l)−al+bj
+�
+j + n + 1
+2
+�2 ≤
+1
+�
+n + 1
+2
+�2
+�
+0≤j≤l<∞
+e(−γ+b)j−(γ+a)l <
+const
+�
+n + 1
+2
+�2
+which clearly goes to zero as n → ∞.
+In the case of C2, we get
+SC2 =
+�
+0≤j≤l
+�
+l + 1
+2
+�2 · · ·
+�
+l + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2 · e−(γ+a)n+(−γ+b)j−(γ+a)l
+�
+j + n + 1
+2
+�2
+.
+Letting l = k + j, the sum becomes
+SC2 =
+�
+j,k≥0
+�
+k + j + 1
+2
+�2 · · ·
+�
+k + j + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2
+· e−(γ+a)n+(−γ+b)j−(γ+a)(k+j)
+�
+j + n + 1
+2
+�2
+.
+Implementing Lemma 4.2 we have
+SC2 ≤ (2n + 1)e−(γ+a)n
+�
+1 − e− γ+a
+2
+�2n+1 → 0 as n → ∞ .
+In sub-region C3 we have:
+SC3 =
+�
+0≤l≤j≤l+n
+�
+l + 1
+2
+�2 · · ·
+�
+l + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2 · e−(γ−b)j−(γ+a)(l+n)
+�
+j + n + 1
+2
+�2
+Notice that in this region we again have that
+�
+l + 1
+2
+�2 · · ·
+�
+l + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2 ≤ 1.
+
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS
+11
+Thus, by overestimating, we have that
+SC3 ≤
+�
+0≤l≤j≤∞
+e−(γ−b)j−(γ+a)(l+n)
+�
+j + n + 1
+2
+�2
+≤
+1
+�
+n + 1
+2
+�2
+�
+0≤l≤j≤∞
+e−(γ−b)j−(γ+a)(l+n)
+which goes to zero as n → ∞.
+Region D: l ≤ j ≤ 0.
+First we map j �→ −j and l �→ −l and the sum becomes
+SD =
+�
+0≤j≤l
+�
+l − 1
+2
+�2 · · ·
+�
+l − n + 1
+2
+�2
+�
+j − 1
+2
+�2 · · ·
+�
+j − n + 1
+2
+�2 · eγj−γl−a|l|+b|j|
+�
+j − n − 1
+2
+�2 .
+Like with region C, we split this into three sub-regions: D1: 0 ≤ j ≤ n ≤ l, D2: n ≤ j ≤ l,
+and D3: 0 ≤ j ≤ l ≤ n.
+In sub-region D1, we map l �→ l + n which yields
+SD1 =
+�
+0≤j≤n,l≥0
+�
+l + 1
+2
+�2 · · ·
+�
+l + n − 1
+2
+�2
+�
+j − 1
+2
+�2 · · ·
+�
+j − n + 1
+2
+�2 · e−(γ+a)ne(γ+b)j−(γ+a)l
+�
+j − n − 1
+2
+�2
+.
+Multiplying and dividing the sum by ( 1
+2)2 · · · (n − 1
+2)2 and using Lemma 4.2 we get
+SD1 ≤ (2n + 1)e−(γ+a)n
+�
+1 − e− γ+a
+2
+�2n+1
+�
+0≤j≤n
+� 1
+2
+�2 · · ·
+�
+n − 1
+2
+�2
+�
+j − 1
+2
+�2 · · ·
+�
+j − n + 1
+2
+�2 ·
+e(γ+b)j
+�
+j − n − 1
+2
+�2 .
+The second inequality in Lemma 4.1 implies that
+SD1 ≤ (2n + 1)e−(γ+a)n
+�
+1 − e− γ+a
+2
+�2n+1
+�
+0≤j≤n
+�
+2n
+2j
+�
+2je(γ+b)j
+�
+j − n − 1
+2
+�2
+≤ (2n + 1)e−(γ+a)n
+�
+1 − e− γ+a
+2
+�2n+1
+�
+0≤j≤n
+�
+2n
+j
+�
+e( γ+b
+2 )j
+≤ n(2n + 1)
+�
+e− γ+a
+2 + e− (a−b)
+2
+1 − e− γ+a
+2
+�2n
+.
+The conditions on a, b and γ imply that the right hand side of the above inequality goes to
+0 as n → ∞ and thus SD1 → 0 as n → ∞.
+In sub-region D2, we map l �→ l + n and j �→ j + n to get
+SD2 =
+�
+0≤j≤l
+�
+l + 1
+2
+�2 · · ·
+�
+l + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2 · e−(γ+a)(l+n)+(γ+b)(j+n)
+�
+j − 1
+2
+�2
+.
+
+12
+KLIMEK, MCBRIDE, AND SAKAI
+Writing l = k + j and using Lemma 4.2 we obtain:
+SD2 = e−(a−b)n �
+0≤j,k
+�
+j + k + 1
+2
+�2 · · ·
+�
+j + k + n − 1
+2
+�2
+�
+j + 1
+2
+�2 · · ·
+�
+j + n − 1
+2
+�2
+· e−(a−b)j−(γ+a)k
+�
+j − 1
+2
+�2
+≤ (2n + 1)e−(a−b)n
+�
+1 − e− γ+a
+2
+�2n+1
+�
+0≤j
+e−(a−b)j
+�
+j − 1
+2
+�2 < ∞ .
+Like in sub-region D1, the conditions on a, b and γ imply the last term in the above inequality
+goes to zero as n → ∞ and hence SD2 goes to zero as n → ∞.
+Finally in the sub-region D3, by multiplying and dividing by ( 1
+2)2 · · · (n − 1
+2)2, we have
+that:
+SD3 =
+�
+0≤j≤l≤n
+qn(j)
+qn(l) · e−(γ+a)l+(γ+b)j
+�
+j − n − 1
+2
+�2 .
+Observe in this region we have that n + 1
+2 − j > n + 1
+3 − j. Using this observation and the
+third inequality in Lemma 4.1 we have
+SD3 ≤
+�
+0≤j≤l
+�
+2l
+2j
+�
+· e−( γ+a
+2 )2l+( γ+b
+2 )2j
+�1
+2 · 2j − n − 1
+3
+�2 ≤
+�
+0≤j≤l≤2n
+�
+l
+j
+�
+· e−( γ+a
+2 )l+( γ+b
+2 )j
+� j
+2 − n − 1
+3
+�2
+=
+�
+l≥0
+�
+l
+�
+j=0
+�
+l
+j
+�
+e( γ+b
+2 )j
+� j
+2 − n − 1
+3
+�2
+�
+e−( γ+a
+2 )l =
+�
+l≥0
+1
+� l
+2 − n − 1
+3
+�2
+�
+e−( γ+a
+2 ) + e−( a−b
+2 )�l
+where the last sum is finite.
+Thus by the Lebesgue Dominated Convergence Theorem,
+SD2 → 0 as n → ∞. This completes the proof.
+□
+Remark. Since a bounded perturbation of an operator with compact parametrices also has
+compact parametrices, see Appendix of [4], it follows for example that, if β(l) = β∞l + ˜β(l)
+and α(l) = e−γ/2β∞l + ˜α(l), where β∞ is a nonzero constant, ˜α(l) and ˜β(l) are bounded,
+then the corresponding operator D has compact parametrices. This observation substantially
+increases the class of covariant implementations with compact parametrices.
+4.2. Proofs of Lemmas.
+Proof. (of Lemma 4.1) Notice that
+qn(1) =
+� 1
+2
+�2 �3
+2
+�2 · · ·
+�
+n − 3
+2
+�2 �
+n − 1
+2
+�2
+�1
+2
+�2 � 1
+2
+�2 �3
+2
+�2 · · ·
+�
+n − 5
+2
+�2 �
+n − 3
+2
+�2 = (2n − 1)2
+12
+.
+Similarly
+qn(2) = (2n − 3)2(2n − 1)2
+12 · 32
+.
+It follows by induction that
+qn(l) = qn(l − 1)(2n − 2l + 1)2
+(2l − 1)2
+= ((2n − 2l + 1) · · ·(2n − 3)(2n − 1))2
+(1 · 3 · 5 · · ·(2l − 1))2
+.
+(4.1)
+
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS
+13
+For the first inequality, notice that from the inductive formula (4.1) we see that qn(l) as a
+function of l, 0 ≤ l ≤ n, first increases and then decreases, so the minimum of it occurs at
+the endpoints. This means that qn(l) ≥ qn(0) = qn(n) = 1, yielding the first inequality.
+To prove the second inequality notice that
+qn(l) = ((2n − 2l + 1) · · ·(2n − 3)(2n − 1))2
+(1 · 3 · 5 · · ·(2l − 1))2
+≤ (2n − 2l + 1)(2n − 2l + 2) · · · (2n − 1)(2n)
+1 · 2 · 3 · · ·(2l − 2)(2l − 1)
+=
+(2n)!
+(2n − 1)!(2n − 2l)!
+=
+2l(2n)!
+(2l)!(2n − 2l)! = 2l
+�
+2n
+2l
+�
+.
+To prove the final inequality we estimate as follows:
+qn(j)
+qn(l) =
+�
+l − 1
+2
+�2 · · ·
+�
+l − n + 1
+2
+�2
+�
+j − 1
+2
+�2 · · ·
+�
+j − n + 1
+2
+�2 =
+(2j + 1)2 · · · (2l − 1)2
+(2n − 2l + 1)2 · · · (2n − 2j + 1)2
+≤
+(2j + 1)(2j + 2) · · · (2l − 1)(2l)
+(2n − 2l)(2n − 2l + 1) · · ·(2n − 2j + 1) = (2l)!
+(2j)! · (2n − 2l − 1)!
+(2n − 2j − 1)!
+=
+�
+2l
+2j
+�
+�
+2n − 2j − 1
+2n − 2l − 1
+� ≤
+�
+2l
+2j
+�
+.
+□
+To prove Lemma 4.2, we need the following additional step.
+Lemma 4.4. For a nonnegative integer j, define the following sum:
+Im(j) =
+�
+k≥0
+(k + 2j + 1) · · ·(k + 2j + m)e− γ
+2 k
+m!
+for m > 0 and I0 =
+�
+1 − e− γ
+2
+�−1
+.
+Then, for m ≥ 0, we have:
+Im(j) ≤
+1
+�
+1 − e− γ
+2 �m+1 · (2j + 1) · · ·(2j + m)
+m!
+.
+Proof. Notice Im(j) satisfies the following reduction formula:
+Im(j) = (2j + 1) · · · (2j + m)
+m!
++ e− γ
+2 Im(j) +
+�
+k≥1
+(k + 2j + 1) · · ·(k + 2j + m − 1)e− γ
+2 k
+(m − 1)!
+= (2j)(2j + 1) · · ·(2j + m − 1)
+m!
++ e− γ
+2 Im(j) + Im−1(j) .
+This implies that Im(j) satisfies the following recurrence relation:
+�
+1 − e− γ
+2
+�
+Im(j) − Im−1(j) =
+�
+2j + m − 1
+2j − 1
+�
+.
+
+14
+KLIMEK, MCBRIDE, AND SAKAI
+This can be solved recursively which yields
+Im(j) =
+1
+�
+1 − e− γ
+2 �m+1
+m
+�
+r=0
+�
+2j + r − 1
+r
+� �
+1 − e− γ
+2
+�r
+.
+Since 1 − e−γ/2 ≤ 1 we get
+Im(j) ≤
+1
+�
+1 − e− γ
+2 �m+1
+m
+�
+r=0
+�
+2j + r − 1
+r
+�
+.
+By parallel summation we have
+m
+�
+r=0
+�
+2j + r − 1
+r
+�
+=
+�
+2j + m
+m
+�
+= (2j + 1) · · ·(2j + m)
+m!
+and thus the result follows.
+□
+Proof. (of Lemma 4.2) Observe the following inequality
+Jn(j) =
+�
+k≥0
+(2k + 2j + 1)2 · · · (2k + 2j + 2n − 1)2
+(2j + 1)2 · · · (2j + 2n − 1)2
+e− γ
+2 ·2k
+≤
+�
+k≥0
+(2k + 2j + 1)(2k + 2j + 2) · · ·(2k + 2j + 2n)
+(2j + 1)2 · · · (2j + 2n − 1)2
+e− γ
+2 ·2k.
+Overestimating by adding odd 2k + 1 terms we get:
+Jn(j) ≤
+(2n)!
+(2j + 1)2 · · · (2j + 2n − 1)2
+�
+k≥0
+(k + 2j + 1) · · ·(k + 2j + 2n)
+(2n)!
+e− γ
+2 k
+=
+(2n)!
+(2j + 1)2 · · · (2j + 2n − 1)2I2n(j),
+where I2n(j) is defined in Lemma 4.4. By implementing Lemma 4.4 we arrive at
+Jn(j) ≤
+(2n)!
+(2j + 1)2 · · · (2j + 2n − 1)2 · (2j + 1) · · ·(2j + 2n)
+(2n)!
+·
+1
+�
+1 − e− γ
+2 �2n+1
+=
+(2j + 2)(2j + 4) · · · (2j + 2n)
+(2j + 1)(2j + 3) · · ·(2j + 2n − 1) ·
+1
+�
+1 − e− γ
+2 �2n+1
+≤ 2j + 2n
+2j + 1 ·
+1
+�
+1 − e− γ
+2 �2n+1 ≤
+2n + 1
+�
+1 − e− γ
+2 �2n+1 .
+□
+References
+[1] Connes, A., Non-Commutative Differential Geometry, Academic Press, 1994.
+[2] Forsyth, I., Mesland, B., Rennie, A., Dense domains, symmetric operators and spectral triples, New
+York J. Math., 20, 1001 - 1020, 2014.
+[3] Klimek, S. and McBride, M., D-bar Operators on Quantum Domains. Math. Phys. Anal. Geom., 13,
+357 - 390, 2010.
+
+IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS
+15
+[4] Klimek, S., McBride, M., Rathnayake, S., Sakai, and Wang, H., Derivations and Spectral Triples on
+Quantum Domains I: Quantum Disk, SIGMA, 013, 1 - 26, 2017.
+[5] Klimek, S., McBride, M., and Rathnayake, S., Derivations and Spectral Triples on Quantum Domains
+II: Quantum Annulus, Sci. Chi. Math., 12, 2463 - 2486, 2019.
+[6] Klimek, S., McBride, M., and Peoples, J.W., A Note on Spectral Triples on the Quantum Disk, SIGMA,
+015, 1 - 8, 2019.
+[7] Klimek, S., McBride, M., and Peoples, J.W., Noncommutative Geometry of the Quantum Disk, Ann.
+Funct. Analysis, 13, 53, 2022
+Department of Mathematical Sciences, Indiana University-Purdue University Indianapo-
+lis, 402 N. Blackford St., Indianapolis, IN 46202, U.S.A.
+Email address: [email protected]
+Department of Mathematics and Statistics, Mississippi State University, 175 President’s
+Cir., Mississippi State, MS 39762, U.S.A.
+Email address: [email protected]
+Department of Mathematical Sciences, Indiana University-Purdue University Indianapo-
+lis, 402 N. Blackford St., Indianapolis, IN 46202, U.S.A.
+Email address: [email protected]
+

A9FAT4oBgHgl3EQfrx7P/content/tmp_files/load_file.txt

@@ -0,0 +1,359 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf,len=358
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='08655v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='OA]  20 Jan 2023  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM  ANNULUS  SLAWOMIR KLIMEK, MATT MCBRIDE, AND KAORU SAKAI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We construct compact parametrix implementations of covariant derivations on  the quantum annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Introduction  The goal of this paper is to provide simple examples of Dirac type operators on noncom-  mutative compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We study analogs of d-bar operators on the quantum annulus  using only inherent geometrical structures: rotations, invariant states, covariant derivations  and their implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' In our previous paper [3], we constructed similar d-bar type  operators on the quantum annulus using APS-type boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The class of oper-  ators was designed to mimic the classical Atiyah-Patodi-Singer theory and is different, less  geometrical than the class studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The main outcome, as in the past paper,  is that we show that our quantum d-bar type operators have compact parametrices, like  elliptic differential operators on compact manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Our another paper [5] on quantum annulus, following [4], contains a description of un-  bounded derivations, covariant with respect to a natural rotation, and their implementations  in Hilbert spaces obtained from the GNS construction with respect to invariant states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' It  turned out that no such implementation in any GNS Hilbert space for a faithful, normal,  invariant state has compact parametrices for a large class of boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' However,  as demonstrated in [6], if we relax the concept of an implementation by allowing operators  to act between different Hilbert spaces, then there is an interesting class of examples of  quantum d-bar operators with compact parametrices that can be constructed this way for  the case of the quantum disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' It is the purpose of this paper to extend those ideas to the  quantum annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Spectral triples are a key tool in noncommutative geometry [1], as they allow using ana-  lytical methods in studying quantum spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Since compact parametrix property is a part  of the spectral triples conditions, our papers [4] and [5] demonstrate, using analytic tech-  niques, that spectral triples, in general, cannot be constructed on the quantum disk and  the quantum annulus using implementations of covariant derivations in GNS Hilbert spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Another, topological reason was pointed out in [7], in the case of the quantum disk T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Namely, the pull-back map in K-Homology K0(C(S1)) → K0(T ) is an isomorphism and so  the restriction map K0(T ) → K0(K) is a zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Consequently, any spectral triple over  the Toeplitz algebra, when restricted to the ideal of compact operators K should be trivial in  K-Homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' However, it is easy to compute that implementations of covariant derivations  Date: January 23, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  1  2  KLIMEK, MCBRIDE, AND SAKAI  pair nontrivially with a minimal projection in K, and hence they cannot lead to spectral  triples over T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Similar arguments seem to also apply to the quantum annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Additionally, as pointed out in [2], there are fundamental reasons why APS boundary  conditions are not compatible with spectral triples even in classical geometry for algebras  of functions which are non-constant on the boundary, as the corresponding domains of the  Dirac-type operators are not preserved by the representations of the algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  In [6], the authors claimed to construct an even spectral triple over the quantum disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Due  to technicalities in the definition of an implementation of a derivation, however, this was not  true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We clarify the generalized concept of implementations of unbounded derivations and  when they lead to spectral triples in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' In Section 3 we establish the notation  and review results from [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The main result, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='3, is proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' It states  that, for a class of exponential coefficients, the operator D defined in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='6) as a  suitable Hilbert spaces implementation of a covariant derivation δ of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2), in the quantum  annulus algebra A, see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1), has a compact parametrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' in fact the inverse of D is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Implementations of Unbounded Derivations  Let A be a C∗-algebra, A ⊆ A a dense ∗-subalgebra and δ : A �→ A a derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Suppose  that H1 and H2 are Hilbert spaces carrying representations of A and denoted π1 and π2  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  The following is a natural concept of an implementation of a derivation in  A between two Hilbert spaces, generalizing the usual notion of an implementation of an  unbounded derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  An implementation of δ between H1 and H2 consists of the following:  A dense subspace dom(D) ⊆ H1  An implementing operator D : dom(D) → H2  An intertwinner i : dom(D) → H2  such that  (1) ∀a ∈ A, ∀x ∈ dom(D)  π1(a)x ∈ dom(D)  (2) ∀a ∈ A, ∀x ∈ dom(D)  i(π1(a)x) = π2(a)i(x)  (3) ∀a ∈ A, ∀x ∈ dom(D)  Dπ1(a)x − π2(a)Dx = π2(δ(a))i(x)  A special case of the above definition is when H1 = H2 = H, π1 = π2 = π and i is  the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Then the second condition above is obviously satisfied, while the third  condition can be written as  (Dπ(a) − π(a)D)x = π(δ(a))x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  This coincides with the usual concept of an unbounded implementation of a derivation as a  commutator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Recall that a closed operator D is called a Fredholm operator if there are bounded op-  erators Q1 and Q2 such that Q1D − I and DQ2 − I are compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The operators Q1 and  Q2 are called left and right parametrices respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We say that a Fredholm operator D  has compact parametrices if at least one (and consequently both) of the parametrices Q1  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS  3  and Q2 is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' More on general properties of operators with compact parametrices can  be found in the appendix of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We also say that an implementation (dom(D), D, i) has  compact parametrices if the closure of the operator D has compact parametrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Under additional conditions an implementation of a derivation can lead to an even spectral  triple over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Namely, defining H = H1  � H2, with grading Γ  ��  H1 = 1 and Γ  ��  H2 = −1 and a  representation π : A → B(H) of A in H given by the formula:  π(a) = (π1(a), π2(a)),  and also defining a generally unbounded operator D in H by:  D =  �  0  D  D∗  0  �  ,  we see that π(a) are even and D is odd with respect to grading Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' If D is a self-adjoint  operator with compact parametrix and additionally the intertwinner i is bounded, then the  conditions in the definition of a derivation implementation imply that π(a) preserve the  domain of D for all a ∈ A and the commutator [D, π(a)] is bounded as can be seen by  a simple calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Consequently, under those additional conditions, we obtain an even  spectral triple over A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  A very natural class of implementations of derivations can be obtained from GNS rep-  resentations in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Suppose τ1, τ2 are faithful states on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Let H1, H2 be  the corresponding GNS Hilbert spaces, obtained by completing A with respect to the inner  products  (a, b)i = τi(a∗b),  i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Because we assume that the states are faithful, A sits densely in H1, H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' More precisely,  there are injective continuous linear maps φ1 : A → H1, φ1 : A → H1 with dense ranges  embedding A into H1, H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  The Hilbert spaces H1, H2 carry natural representations of A given by left multiplication;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  for a, b ∈ A we have  πi(a)φi(b) = φi(ab).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Suppose as before that we have a dense ∗-subalgebra A ⊆ A and a derivation δ : A �→ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Then we have the following implementation of δ between H1 and H2:  dom(D) := φ1(A) ⊆ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' If x ∈ dom(D) then we write x = φ1(b) for some b ∈ A,  notation we use in the formulas below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  D(x) := φ2(δ(b)),  i(x) := φ2(b)  It is a matter of straightforward calculations to verify that indeed the three conditions of the  definition are satisfied and the above defines an implementation of δ between H1 and H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Quantum Annulus Preliminaries  We review the notation and basic concepts from [5] below and, in a number of places, we  use the results contained in that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  4  KLIMEK, MCBRIDE, AND SAKAI  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The Quantum Annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Let {El}l∈Z be the canonical basis for ℓ2(Z) and V be the  bilateral shift defined by  V El = El+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Notice that V is a unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Let L be the diagonal label operator defined by  LEl = lEl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  It follows from the functional calculus that given a function a : Z → C, we have  a(L)El = a(l)El .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  These are precisely the operators which are diagonal with respect to {El}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The operators  (L, V ) serve as noncommutative polar coordinates, and they satisfy the following commuta-  tion relation:  LV = V (L + I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Let c(Z) be the set of a(l), as above, which are convergent as l → ±∞ and let c++  00 (Z) be the  set of all eventually constant functions, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' functions a(l) such that there exists a l0 where  a(l) is a constant for l ≥ l0 and also is a possibly different constant for l ≤ −l0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Let A be the C∗-algebra generated by V and a(L), that is:  A = C∗(V, a(L) : a(l) ∈ c(Z))  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1)  This algebra is called the quantum annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The smallest reasonable domain of derivations  in A is the following dense ∗-subalgebra of A:  A =  �  a =  �  n∈Z  V nan(L) : an(l) ∈ c++  00 (Z), finite sums  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Derivations in the Quantum Annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Let ρθ : A → A, 0 ≤ θ < 2π, be a one  parameter group of automorphisms of A defined by:  ρθ(a) = eiθLae−iθL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Since ρθ(a(L)) = a(L), ρθ(V ) = eiθV and consequently ρθ(V −1) = e−iθV −1, the auto-  morphisms ρθ are well defined on A and they preserve A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' By Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2 in [5], any  densely-defined derivation δ : A → A, covariant with respect to ρθ that is  ρθ(δ(a)) = eiθδ(ρθ(a)) ,  is of the following form:  δ(a) = [Uβ(L), a]  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2)  where {β(l + 1) − β(l)} ∈ c(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We use notation:  lim  l→±∞(β(l + 1) − β(l)) := β±∞,  and below we only consider covariant derivations with β±∞ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' It follows that there are  constants c1 and c2 so that  c1(|l| + 1) ≤ |β(l)| ≤ c2(|l| + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='3)  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS  5  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Covariant Implementations on the Quantum Annulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Here we consider covari-  ant implementations of derivations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We begin by introducing the following family of  states τw : A → C on A, defined by  τw(a) = tr(w(L)a) ,  where w(l) > 0 for all l ∈ Z and  �  l∈Z  w(l) = 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  As a result of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='3 in [5], τw are precisely the ρθ-invariant, normal, faithful states  on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Let Hw be the Hilbert space obtained by Gelfand-Naimark-Segal (GNS) construction  on A using state τw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Since the state is faithful, Hw is the completion of A with respect to  the inner product given by  ⟨a, b⟩w = τw(a∗b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  A simple calculation leads to the following precise description:  Hw =  �  f =  �  n∈Z  V nfn(L) : ∥f∥2  w =  �  n∈Z  �  l∈Z  w(l)|fn(l)|2 < ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='4)  With this identification we naturally have A ⊆ Hw so the inclusion maps φw : A → Hw  are the identity maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Notice also that A is dense in Hw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The GNS representation map  πw : A → B(Hw) is given by left-hand multiplication:  πw(a)f = af .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Define a one parameter group of unitary operators V w  θ : Hw → Hw via the formula:  V w  θ f =  �  n∈Z  V neinθfn(L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  An immediate calculation shows that:  πw(ρθ(a)) = V w  θ πw(a)(V w  θ )−1 ,  and therefore the operators V w  θ are implementing the one parameter group of automorphisms  ρθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Consider an additional weight, w′(l), possibly different from w(l), satisfying the same  conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Proceeding like in the previous section, we set  dom(D) := A ⊂ Hw ,  and choose for an implementing operator  i : dom(D) = A → A ⊂ Hw′  to be the identity operator a �→ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Clearly, the first two properties of an implementation are  satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We say that an operator D : Hw ⊇ A → Hw′ defines a covariant implementation  of the covariant derivation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2) if for every a ∈ A, and for every f ∈ A considered as an  element of both Hw and Hw′, we have:  Dπw(a)f − πw′(a)Df = πw′(δ(a))f ,  and, additionally, D satisfies:  V w′  θ D(V w  θ )−1f = eiθDf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  6  KLIMEK, MCBRIDE, AND SAKAI  Proceeding as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2 in [5] shows the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' There exists a sequence {α(l)} satisfying  �  l∈Z  |β(l) − α(l)|2w′(l) < ∞  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='5)  such that any covariant implementation D : Hw ⊇ A → Hw′ is of the form:  Df = V β(L)f − fV α(L) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='6)  Conversely, for any {α(l)} satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='5), the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='6) defines a covariant implemen-  tation D : Hw ⊇ A → Hw′ of the derivation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  We assume below that for every l we have:  α(l), β(l) ̸= 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  It is convenient, like in [4] and [5], to write  α(l) = β(l)µ(l + 1)  µ(l)  for some sequence {µ(l)} such that µ(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Fourier Decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' To further analyze the operator D of formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='6), we can  decompose it into a Fourier series and study its Fourier components which are operators  acting between weighted ℓ2-spaces, defined as follows:  ℓ2  w =  �  {f(l)}l∈Z :  �  l∈Z  |f(l)|2w(l) < ∞  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  We have the following decomposition proposition:  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Let f ∈ dom(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Then  Df =  �  n∈Z  V n+1(Dnfn)(L) ,  where Dn : ℓ2  w ⊇ c++  00 (Z) → ℓ2  w′ and is given by the following formula:  (Dnh)(l) = β(l + n)h(l) − β(l)µ(l + 1)  µ(l)  h(l + 1)  for some h ∈ c++  00 (Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The proof follows by writing f ∈ dom(D) as its Fourier series, applying D to it and  using the commutation relation LV = V (L + I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  □  In what follows, the purpose is to choose the parameters such that D has a compact  parametrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS  7  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Parametrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Next we study a formal candidate for a parametrix of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The closure of  D, defined as above on c++  00 (Z), will be denoted by ¯D while its closure defined on the space  c00(Z) of eventually zero functions will be denoted by ¯D00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Also notice that D preserves  c00(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We have the following simple observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' If {α(l)} satisfies (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='5) then ¯D = ¯D00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Notice that c00(Z) ⊂ dom(D) is a co-dimension 2 subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Thus, it is enough to  verify that 1 and the characteristic function of Z≥0 are in the domain of ¯D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Approximating  1 by characteristic functions χN of sets −N ≤ l ≤ N, χN ∈ c00(Z), we see that D(χN)  converges in ℓ2  w to D(1), which is in ℓ2  w by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='5), implying that 1 is in the closure of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The  characteristic function of Z≥0 is in the domain of ¯D by the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  □  Let Qn be given by the following formula:  (Qng)(l) =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∞  �  j=l  �l+n−1  k=l  β(k)  �j+n  k=j β(k)  µ(j)  µ(l) g(j)  n ≥ 0  ∞  �  j=l  �j−1  k=j+n+1 β(k)  �l−1  k=l+n β(k)  µ(j)  µ(l) g(j)  n < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  This expression was obtained by inverting Dn using techniques similar to the calculations in  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='12 in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Notice that we have:  Qn : c00(Z) → dom(D),  since the sums in the definitions of Qn are finite for g ∈ c00(Z) and the outcomes are  eventually constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Relations between Dn and Qn are explained in the following statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' For every f ∈ c00(Z) we have:  DnQnf = f  and QnDnf = f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' The formulas follow from straightforward calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  □  From this proposition we can formally define the inverse for D to be Q = �  n∈Z Qn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' For  this to be well-defined, the series needs to converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' In fact, once we verify that Q is bounded,  the previous two propositions imply that Q is the inverse of ¯D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Results  For the remainder of this section we assume that β(k) = k + 1  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Moreover we only consider  the special choices of the weights {w(l)}, {w′(l)} and the choice for {µ(l)} namely:  w(l) = e−a|l|,  w′(l) = e−b|l|,  and µ(l) = e−(γl)/2, for a, b, γ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  It should be noted that with these specific choices α(l) = e−γ/2β(l) and the conditions  in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1 are trivially satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  By a simple perturbative argument the results  below are valid for a much larger class of coefficients, see the remark at the end of the next  subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  8  KLIMEK, MCBRIDE, AND SAKAI  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Compactness of Parametrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We show that Qn are Hilbert-Schmidt operators for  every n and verify that their respective Hilbert-Schmidt norms go to zero as n goes to infinity,  implying that Q is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' To prove this we need a few helper lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We  postpone proofs of those lemmas until the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' For 0 ≤ l ≤ n, define the following product:  qn(l) =  � 1  2  �2 � 3  2  �2 · · ·  �  n − 1  2  �2  �  l − 1  2  �2 �  l − 3  2  �2 · · ·  �  l − n + 1  2  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Then we have the identity:  qn(l) = ((2n − 2l + 1) · · ·(2n − 3)(2n − 1))2  (1 · 3 · 5 · · ·(2l − 1))2  for every natural number n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Moreover, qn(l) satisfies the following three estimates:  (1) qn(l) ≥ 1,  (2) qn(l) ≤ 2l  �  2n  2l  �  ,  (3) qn(j)  qn(l) ≤  �  2n  2l  �  for 0 ≤ j ≤ l .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' For a nonnegative integer j, define the following sum:  Jn(j) =  �  k≥0  �  k + j + 1  2  �2 · · ·  �  k + j + n − 1  2  �2 e−γk  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Then for n ≥ 0  Jn(j) ≤  2n + 1  �  1 − e− γ  2 �2n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  The following is the main technical result of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Suppose that γ > a > b and that  exp  �  −(a − b)  2  �  + exp  �  −γ + a  2  �  < 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Then Qn : ℓ2  w′ → ℓ2  w is a Hilbert-Schmidt operator for every n ∈ Z and ∥Qn∥HS → 0 as  n → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Consequently, Q = �  n Qn is the inverse of ¯D and is a compact operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Notice that formula for Qn shows that it is an integral operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' by direct  calculation we can compute the Hilbert-Schmidt norms:  ∥Qn∥2  HS =  \uf8f1  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f2  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f4  \uf8f3  ∞  �  j=l  �l+n−1  k=l  |β(k)|2  �j+n  k=j |β(k)|2 · |µ(j)|2  |µ(l)|2 · w(l)  w′(j)  n ≥ 0  ∞  �  j=l  �j−1  k=j+n+1 |β(k)|2  �l−1  k=l+n |β(k)|2  |µ(j)|2  |µ(l)|2 · w(l)  w′(j)  n < 0  Since the choice of β’s are linear,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' the following ratios:  ����  β(l) · · ·β(l + n − 1)  β(l + n) · · ·β(l − 1)  ����  2  and  ����  β(j + n + 1) · · ·β(j − 1)  β(l + n) · · ·β(l − 1)  ����  2  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS  9  are ratios of polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Thus by our choice of exponential µ’s, w’s and w′’s, ∥Qn∥HS exists  if and only if  �  j≥l  ����  µ(j)  µ(l)  ����  2  w(l)  w′(j) < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  However, this easily follows if γ > a > b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Thus the Hilbert-Schmidt norm of Qn exists for  all n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' It remains to show that those norms go to zero as n → ±∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' We only need to  study the case for n ≥ 0 as if n < 0 then by doing a change of variables of j �→ −l, l �→ −j  and n �→ −n − 1 we are back in the n ≥ 0 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Thus, we only need to estimate the sum:  ∥Qn∥2  HS =  �  j≥l  �  l + 1  2  �2 · · ·  �  l + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2 · e−γ(j−l)−a|l|+b|j|  �  j + n + 1  2  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  The sum is over all j ≥ l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' It splits into sums over four main regions which we will further  subdivide as illustrated in the picture below:  We have  ∥Qn∥2  HS ≤ SA + SB + SC + SD,  as for simplicity of estimations we let the regions overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Here the regions are: region A:  j ≥ l ≥ 0, region B: j ≥ −l ≥ 0, l ≤ 0, region C: −l ≥ j ≥ 0, l ≤ 0, and region D: l ≥ j ≥ 0,  which we will handle separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' In our estimates, we double count the boundaries in some  places for convenience of different estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Regions A and B: Notice the following observation: if −j ≤ l ≤ j for j ≥ 0, then for  any r > 0 we have that (l + r)2 ≤ (j + r)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Using this fact, we can estimate:  �  l + 1  2  �2 · · ·  �  l + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  B  A  C3  C2  C1  D3  D1  D210  KLIMEK, MCBRIDE, AND SAKAI  It follows that we have  SA ≤  �  j≥l≥0  e−γ(j−l)−al+bj  �  j + n + 1  2  �2 → 0 as n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Changing l → −l we get exactly the same estimate for SB, and so SB → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Region C: 0 ≤ j ≤ −l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  First we map l �→ −l, then we split this region into two sub-regions: C1: 0 ≤ j ≤ l ≤ n  and the complement of C1: 0 ≤ j ≤ l, l ≥ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' In the second region we make a change of  variables of l′ = l − n, then replace l′ with l and we further get two additional sub-regions:  C2: 0 ≤ j ≤ l and C3: 0 ≤ l ≤ j ≤ l + n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  First in sub-region C1: since l ≤ n, using the first inequality from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1, we have  SC1 =  �  0≤j≤l≤n  �  l − 1  2  �2 · · ·  �  l − n + 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2 · e−γ(j+l)−al+bj  �  j + n + 1  2  �2  ≤  �  0≤j≤l≤n  �  l − 1  2  �2 · · ·  �  l − n + 1  2  �2  �1  2  �2 · · ·  �  n − 1  2  �2  e−γ(j+l)−al+bj  �  j + n + 1  2  �2  ≤  �  0≤j≤l≤n  e−γ(j+l)−al+bj  �  j + n + 1  2  �2 ≤  1  �  n + 1  2  �2  �  0≤j≤l<∞  e(−γ+b)j−(γ+a)l <  const  �  n + 1  2  �2  which clearly goes to zero as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  In the case of C2, we get  SC2 =  �  0≤j≤l  �  l + 1  2  �2 · · ·  �  l + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2 · e−(γ+a)n+(−γ+b)j−(γ+a)l  �  j + n + 1  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Letting l = k + j, the sum becomes  SC2 =  �  j,k≥0  �  k + j + 1  2  �2 · · ·  �  k + j + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2  e−(γ+a)n+(−γ+b)j−(γ+a)(k+j)  �  j + n + 1  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Implementing Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2 we have  SC2 ≤ (2n + 1)e−(γ+a)n  �  1 − e− γ+a  2  �2n+1 → 0 as n → ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  In sub-region C3 we have:  SC3 =  �  0≤l≤j≤l+n  �  l + 1  2  �2 · · ·  �  l + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2 · e−(γ−b)j−(γ+a)(l+n)  �  j + n + 1  2  �2  Notice that in this region we again have that  �  l + 1  2  �2 · · ·  �  l + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS  11  Thus, by overestimating, we have that  SC3 ≤  �  0≤l≤j≤∞  e−(γ−b)j−(γ+a)(l+n)  �  j + n + 1  2  �2  ≤  1  �  n + 1  2  �2  �  0≤l≤j≤∞  e−(γ−b)j−(γ+a)(l+n)  which goes to zero as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Region D: l ≤ j ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  First we map j �→ −j and l �→ −l and the sum becomes  SD =  �  0≤j≤l  �  l − 1  2  �2 · · ·  �  l − n + 1  2  �2  �  j − 1  2  �2 · · ·  �  j − n + 1  2  �2 · eγj−γl−a|l|+b|j|  �  j − n − 1  2  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Like with region C, we split this into three sub-regions: D1: 0 ≤ j ≤ n ≤ l, D2: n ≤ j ≤ l,  and D3: 0 ≤ j ≤ l ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  In sub-region D1, we map l �→ l + n which yields  SD1 =  �  0≤j≤n,l≥0  �  l + 1  2  �2 · · ·  �  l + n − 1  2  �2  �  j − 1  2  �2 · · ·  �  j − n + 1  2  �2 · e−(γ+a)ne(γ+b)j−(γ+a)l  �  j − n − 1  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Multiplying and dividing the sum by ( 1  2)2 · · · (n − 1  2)2 and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2 we get  SD1 ≤ (2n + 1)e−(γ+a)n  �  1 − e− γ+a  2  �2n+1  �  0≤j≤n  � 1  2  �2 · · ·  �  n − 1  2  �2  �  j − 1  2  �2 · · ·  �  j − n + 1  2  �2 ·  e(γ+b)j  �  j − n − 1  2  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  The second inequality in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1 implies that  SD1 ≤ (2n + 1)e−(γ+a)n  �  1 − e− γ+a  2  �2n+1  �  0≤j≤n  �  2n  2j  �  2je(γ+b)j  �  j − n − 1  2  �2  ≤ (2n + 1)e−(γ+a)n  �  1 − e− γ+a  2  �2n+1  �  0≤j≤n  �  2n  j  �  e( γ+b  2 )j  ≤ n(2n + 1)  �  e− γ+a  2 + e− (a−b)  2  1 − e− γ+a  2  �2n  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  The conditions on a, b and γ imply that the right hand side of the above inequality goes to  0 as n → ∞ and thus SD1 → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  In sub-region D2, we map l �→ l + n and j �→ j + n to get  SD2 =  �  0≤j≤l  �  l + 1  2  �2 · · ·  �  l + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2 · e−(γ+a)(l+n)+(γ+b)(j+n)  �  j − 1  2  �2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  12  KLIMEK, MCBRIDE, AND SAKAI  Writing l = k + j and using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2 we obtain:  SD2 = e−(a−b)n �  0≤j,k  �  j + k + 1  2  �2 · · ·  �  j + k + n − 1  2  �2  �  j + 1  2  �2 · · ·  �  j + n − 1  2  �2  e−(a−b)j−(γ+a)k  �  j − 1  2  �2  ≤ (2n + 1)e−(a−b)n  �  1 − e− γ+a  2  �2n+1  �  0≤j  e−(a−b)j  �  j − 1  2  �2 < ∞ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Like in sub-region D1, the conditions on a, b and γ imply the last term in the above inequality  goes to zero as n → ∞ and hence SD2 goes to zero as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Finally in the sub-region D3, by multiplying and dividing by ( 1  2)2 · · · (n − 1  2)2, we have  that:  SD3 =  �  0≤j≤l≤n  qn(j)  qn(l) · e−(γ+a)l+(γ+b)j  �  j − n − 1  2  �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Observe in this region we have that n + 1  2 − j > n + 1  3 − j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Using this observation and the  third inequality in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1 we have  SD3 ≤  �  0≤j≤l  �  2l  2j  �  e−( γ+a  2 )2l+( γ+b  2 )2j  �1  2 · 2j − n − 1  3  �2 ≤  �  0≤j≤l≤2n  �  l  j  �  e−( γ+a  2 )l+( γ+b  2 )j  � j  2 − n − 1  3  �2  =  �  l≥0  �  l  �  j=0  �  l  j  �  e( γ+b  2 )j  � j  2 − n − 1  3  �2  �  e−( γ+a  2 )l =  �  l≥0  1  � l  2 − n − 1  3  �2  �  e−( γ+a  2 ) + e−( a−b  2 )�l  where the last sum is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Thus by the Lebesgue Dominated Convergence Theorem,  SD2 → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  □  Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Since a bounded perturbation of an operator with compact parametrices also has  compact parametrices, see Appendix of [4], it follows for example that, if β(l) = β∞l + ˜β(l)  and α(l) = e−γ/2β∞l + ˜α(l), where β∞ is a nonzero constant, ˜α(l) and ˜β(l) are bounded,  then the corresponding operator D has compact parametrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' This observation substantially  increases the class of covariant implementations with compact parametrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Proofs of Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' (of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1) Notice that  qn(1) =  � 1  2  �2 �3  2  �2 · · ·  �  n − 3  2  �2 �  n − 1  2  �2  �1  2  �2 � 1  2  �2 �3  2  �2 · · ·  �  n − 5  2  �2 �  n − 3  2  �2 = (2n − 1)2  12  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Similarly  qn(2) = (2n − 3)2(2n − 1)2  12 · 32  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  It follows by induction that  qn(l) = qn(l − 1)(2n − 2l + 1)2  (2l − 1)2  = ((2n − 2l + 1) · · ·(2n − 3)(2n − 1))2  (1 · 3 · 5 · · ·(2l − 1))2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1)  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS  13  For the first inequality, notice that from the inductive formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='1) we see that qn(l) as a  function of l, 0 ≤ l ≤ n, first increases and then decreases, so the minimum of it occurs at  the endpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' This means that qn(l) ≥ qn(0) = qn(n) = 1, yielding the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  To prove the second inequality notice that  qn(l) = ((2n − 2l + 1) · · ·(2n − 3)(2n − 1))2  (1 · 3 · 5 · · ·(2l − 1))2  ≤ (2n − 2l + 1)(2n − 2l + 2) · · · (2n − 1)(2n)  1 · 2 · 3 · · ·(2l − 2)(2l − 1)  =  (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (2n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' (2n − 2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  =  2l(2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' (2n − 2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' = 2l  �  2n  2l  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  To prove the final inequality we estimate as follows:  qn(j)  qn(l) =  �  l − 1  2  �2 · · ·  �  l − n + 1  2  �2  �  j − 1  2  �2 · · ·  �  j − n + 1  2  �2 =  (2j + 1)2 · · · (2l − 1)2  (2n − 2l + 1)2 · · · (2n − 2j + 1)2  ≤  (2j + 1)(2j + 2) · · · (2l − 1)(2l)  (2n − 2l)(2n − 2l + 1) · · ·(2n − 2j + 1) = (2l)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (2j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' · (2n − 2l − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (2n − 2j − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  =  �  2l  2j  �  �  2n − 2j − 1  2n − 2l − 1  � ≤  �  2l  2j  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  □  To prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2, we need the following additional step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' For a nonnegative integer j, define the following sum:  Im(j) =  �  k≥0  (k + 2j + 1) · · ·(k + 2j + m)e− γ  2 k  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  for m > 0 and I0 =  �  1 − e− γ  2  �−1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Then, for m ≥ 0, we have:  Im(j) ≤  1  �  1 − e− γ  2 �m+1 · (2j + 1) · · ·(2j + m)  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Notice Im(j) satisfies the following reduction formula:  Im(j) = (2j + 1) · · · (2j + m)  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  + e− γ  2 Im(j) +  �  k≥1  (k + 2j + 1) · · ·(k + 2j + m − 1)e− γ  2 k  (m − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  = (2j)(2j + 1) · · ·(2j + m − 1)  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  + e− γ  2 Im(j) + Im−1(j) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  This implies that Im(j) satisfies the following recurrence relation:  �  1 − e− γ  2  �  Im(j) − Im−1(j) =  �  2j + m − 1  2j − 1  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  14  KLIMEK, MCBRIDE, AND SAKAI  This can be solved recursively which yields  Im(j) =  1  �  1 − e− γ  2 �m+1  m  �  r=0  �  2j + r − 1  r  � �  1 − e− γ  2  �r  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Since 1 − e−γ/2 ≤ 1 we get  Im(j) ≤  1  �  1 − e− γ  2 �m+1  m  �  r=0  �  2j + r − 1  r  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  By parallel summation we have  m  �  r=0  �  2j + r − 1  r  �  =  �  2j + m  m  �  = (2j + 1) · · ·(2j + m)  m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  and thus the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  □  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' (of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='2) Observe the following inequality  Jn(j) =  �  k≥0  (2k + 2j + 1)2 · · · (2k + 2j + 2n − 1)2  (2j + 1)2 · · · (2j + 2n − 1)2  e− γ  2 ·2k  ≤  �  k≥0  (2k + 2j + 1)(2k + 2j + 2) · · ·(2k + 2j + 2n)  (2j + 1)2 · · · (2j + 2n − 1)2  e− γ  2 ·2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Overestimating by adding odd 2k + 1 terms we get:  Jn(j) ≤  (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (2j + 1)2 · · · (2j + 2n − 1)2  �  k≥0  (k + 2j + 1) · · ·(k + 2j + 2n)  (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  e− γ  2 k  =  (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (2j + 1)2 · · · (2j + 2n − 1)2I2n(j),  where I2n(j) is defined in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' By implementing Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='4 we arrive at  Jn(j) ≤  (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  (2j + 1)2 · · · (2j + 2n − 1)2 · (2j + 1) · · ·(2j + 2n)  (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' 1  �  1 − e− γ  2 �2n+1  =  (2j + 2)(2j + 4) · · · (2j + 2n)  (2j + 1)(2j + 3) · · ·(2j + 2n − 1) ·  1  �  1 − e− γ  2 �2n+1  ≤ 2j + 2n  2j + 1 ·  1  �  1 − e− γ  2 �2n+1 ≤  2n + 1  �  1 − e− γ  2 �2n+1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  □  References  [1] Connes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Non-Commutative Differential Geometry, Academic Press, 1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  [2] Forsyth, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Mesland, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Rennie, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Dense domains, symmetric operators and spectral triples, New  York J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', 20, 1001 - 1020, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  [3] Klimek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' and McBride, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', D-bar Operators on Quantum Domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', 13,  357 - 390, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  IMPLEMENTATIONS OF DERIVATIONS ON THE QUANTUM ANNULUS  15  [4] Klimek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', McBride, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Rathnayake, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Sakai, and Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Derivations and Spectral Triples on  Quantum Domains I: Quantum Disk, SIGMA, 013, 1 - 26, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  [5] Klimek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', McBride, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', and Rathnayake, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Derivations and Spectral Triples on Quantum Domains  II: Quantum Annulus, Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Chi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', 12, 2463 - 2486, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  [6] Klimek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', McBride, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', and Peoples, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', A Note on Spectral Triples on the Quantum Disk, SIGMA,  015, 1 - 8, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  [7] Klimek, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', McBride, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', and Peoples, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Noncommutative Geometry of the Quantum Disk, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Analysis, 13, 53, 2022  Department of Mathematical Sciences, Indiana University-Purdue University Indianapo-  lis, 402 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Blackford St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Indianapolis, IN 46202, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Email address: sklimek@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='iupui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='edu  Department of Mathematics and Statistics, Mississippi State University, 175 President’s  Cir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Mississippi State, MS 39762, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='  Email address: mmcbride@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='msstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='edu  Department of Mathematical Sciences, Indiana University-Purdue University Indianapo-  lis, 402 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=' Blackford St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content=', Indianapolis, IN 46202, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
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+page_content='  Email address: ksakai@iupui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}
+page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/A9FAT4oBgHgl3EQfrx7P/content/2301.08655v1.pdf'}

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+Efficiently predicting high resolution mass spectra
+with graph neural networks
+Michael Murphy1∗
+[email protected]
+Stefanie Jegelka1
+[email protected]
+Ernest Fraenkel1
+[email protected]
+Tobias Kind2
+[email protected]
+David Healey2
+[email protected]
+Thomas Butler2
+[email protected]
+1MIT
+2Enveda Biosciences
+January 26, 2023
+Abstract
+Identifying a small molecule from its mass spectrum is the primary open problem in computational
+metabolomics. This is typically cast as information retrieval: an unknown spectrum is matched against
+spectra predicted computationally from a large database of chemical structures.
+However, current
+approaches to spectrum prediction model the output space in ways that force a tradeoff between capturing
+high resolution mass information and tractable learning. We resolve this tradeoff by casting spectrum
+prediction as a mapping from an input molecular graph to a probability distribution over molecular
+formulas. We discover that a large corpus of mass spectra can be closely approximated using a fixed
+vocabulary constituting only 2% of all observed formulas. This enables efficient spectrum prediction using
+an architecture similar to graph classification – GrAFF-MS – achieving significantly lower prediction
+error and orders-of-magnitude faster runtime than state-of-the-art methods.
+1
+Introduction
+The identification of unknown small molecules in complex chemical mixtures is a primary challenge in
+many areas of chemical and biological science. The standard high-throughput approach to small molecule
+identification is tandem mass spectrometry (MS/MS), with diverse applications including metabolomics [1],
+drug discovery [2], clinical diagnostics [3], forensics [4], and environmental monitoring [5].
+The key bottleneck in MS/MS is structural elucidation: given a mass spectrum, we must determine the 2D
+structure of the molecule it represents. This problem is far from solved, and adversely impacts all areas
+of science that use MS/MS. Typically only 2−4% of spectra are identified in untargeted metabolomics
+experiments [6], and a recent competition saw no more than 30% accuracy [7].
+Because MS/MS is a lossy measurement, and existing training sets are small, direct prediction of structures
+from spectra is particularly challenging. Therefore the most common approach is spectral library search, which
+casts the problem as information retrieval [8]: an observed spectrum is queried against a library of spectra with
+known structures. This provides an informative prior, and has the advantage of easy interpretability as the
+entire space of solutions is known. As there are relatively few (104) small molecules with known experimental
+∗The lead author carried out this work as an intern at Enveda Biosciences.
+1
+arXiv:2301.11419v1  [cs.LG]  26 Jan 2023
+
+mass spectra, in spectral library search it is necessary to augment libraries with spectra predicted from large
+databases (106 − 109) of molecular graphs. This motivates the problem of spectrum prediction.
+Spectrum prediction is actively studied in metabolomics and quantum chemistry [9], yet has received little
+attention from the machine learning community. A major challenge in spectrum prediction is modelling of
+the output space: a mass spectrum is a variable-length set of real-valued (m/z, height) tuples, which is not
+straightforward to represent as an output of a deep learning model. The m/z coordinate (mass-to-charge
+ratio) poses particular difficulty: it must be predicted with high precision, as a key strength of MS/MS is the
+ability to distinguish fractional m/z differences on the order of 10−6 representative of different elemental
+composition.
+Existing approaches to spectrum prediction force a tradeoff between capturing high-resolution m/z information
+and tractability of the learning problem. Mass-binning methods [10, 11, 12] represent a spectrum as a fixed-
+length vector by discretizing the m/z axis at regular intervals, discarding valuable information in favor
+of tractable learning. Bond-breaking methods [13, 14] achieve perfect m/z resolution, but do so through
+expensive combinatorial enumeration of substructures.
+This work makes the following contributions:
+• We formulate spectrum prediction as a mapping from a molecular graph to a probability distribution
+over molecular formulas, allowing full resolution predictions without enumerating substructures;
+• We discover most mass spectra can be effectively approximated with a small fixed vocabulary of
+molecular formulas, bypassing the tradeoff between m/z resolution and tractable learning; and
+• We implement an efficient graph neural network architecture, GrAFF-MS, that substantially out-
+performs state-of-the-art in both prediction error and runtime on two canonical mass spectrometry
+datasets.
+2
+Background
+2.1
+Definitions
+We denote vectors x in bold lowercase and matrices X in bold uppercase.
+A molecular graph G = (V, E, a, b) is a minimal description of the structure of a molecule: comprising an
+undirected graph with node set V , edge set E ⊂ V × V , node labels a ∈ [118]V indicating atom number, and
+edge labels b ∈ ({1, 1.5, 2, 3} × {−1, 0, 1})E indicating bond order and chirality.
+A molecular formula f (e.g. C8H10N4O2) describes a multiset of atoms, which we encode as a nonnegative
+integer vector of atom counts in F∗ .= Z118
++ . Formulas may be added and subtracted from one another, and
+inequalities between formulas are taken to hold elementwise. We reserve the symbol P to indicate a formula
+of a precursor ion. The subformulas of P are the set F(P) .= {f ∈ F∗ : f ≤ P}.
+⟨µ, f⟩ ∈ R+ is the theoretical mass of a molecule with formula f, with units of daltons (Da): this is a linear
+combination of the monoisotopic masses of the elements of the periodic table, µ ∈ R118
++ , with multiplicities
+given by f.
+A mass spectrum S is a variable-length set of peaks, each of which is a (m/z, height) tuple (mi, yi) ∈ R2
++. We
+use the notation i ∈ S to index peaks in a spectrum. We assume spectra are normalized, permitting us to
+treat them as probability distributions: �
+i∈S yi = 1. A mass spectrum is implicitly always accompanied by a
+precursor formula P.
+We always assume charge z .= 1, as it is rare for small molecules to acquire more than a single charge.
+2.2
+Tandem mass spectrometry
+A tandem mass spectrometer is a scientific instrument that generates high-throughput experimental signatures
+of the molecules present in a complex mixture. It works by ionizing a chemical sample into a jet of electrically-
+2
+
+charged gas. This gas is electromagnetically filtered to select a population of precursor ions of a specific
+mass-to-charge ratio (m/z) representing a unique molecular structure. Each precursor ion is fragmented by
+collision with molecules of an inert gas. If a collision occurs with sufficient energy, one or more bonds in the
+precursor will break, yielding a charged product ion and one or more uncharged neutral loss molecules. The
+product ion is measured by a detector, which records its m/z up to a small measurement error proportional to
+the m/z times the instrument resolution ϵ. This process is repeated for large numbers of identical precursor
+ions, building up a histogram indexed by m/z. Local maxima in this histogram are termed peaks: ideally,
+each peak represents a unique product ion, with height reflecting its probability of formation. This set of
+peaks constitutes the mass spectrum. A typical mass spectrometry experiment acquires mass spectra for tens
+of thousands of distinct precursors in this manner, with m/zs measured at resolution on the order of 10−6.
+This process is depicted in Figure 1A; we illustrate the relationship between precursor ion, product ion, and
+neutral loss in Figure 1B.
+Figure 1: (A) The workflow of tandem mass spectrometry. A chemical mixture is ionized and filtered to
+isolate precursor ions of m/z = M; these are fragmented into product ions (red) and neutral losses (blue),
+and a detector yields a histogram of product ions indexed by m/z, with measurement error proportional to
+m/z. (B) An example fragmentation of a precursor ion with formula C8H11N4O2. Fragmentation breaks
+bonds, cutting the molecular graph into connected components. The component retaining the charge is the
+product ion; its complement is the neutral loss. The peak at m/z = m represents the product ion; given the
+precursor formula, we can equally specify this peak by its formula C3H4NO2, or the formula of its neutral
+loss C5H7N3.
+2.3
+Structural elucidation
+A tandem mass spectrum of a molecule contains substantial information about its 2D structure. Peak heights
+reflect the propensity of different bonds to break: this can reveal structural differences in compounds, even if
+they have the same molecular formula. Furthermore, the O(10−6) resolution of modern MS/MS can detect
+small characteristic deviations from integrality in the masses of individual chemical elements that arise from
+nuclear physics, allowing detailed information about the molecular formula to be inferred from the spectrum
+with high accuracy (Sec 2.4).
+The task of inferring a 2D structure from a mass spectrum, structural elucidation, is the major open problem
+3
+
+(A)
+M + dM
+M- dM
+m/ z
+(B)
+C8H11N402
+M-m
+N
+N
+C5H7N3
+>m/ z.
+M
+min computational metabolomics, dating back to the 1960s in one of the earliest examples of an expert system
+[15]. Yet structural elucidation remains far from solved: despite half a century of research, algorithmic
+approaches perform only marginally better than manual annotations by expert chemists, with no method
+submitted to the 2022 Critical Analysis of Small Molecule Identification (CASMI) challenge exceeding 30%
+accuracy [7].
+The difficulty of structural elucidation has broad scientific implications: while a modern metabolomics
+experiment routinely detects tens of thousands of spectra, usually only a few percent of these are confidently
+annotated with a structure [6]. Spectral library search, described in the introduction, is the approach to
+structural elucidation preferred in practice by most biologists and chemists [8], and is a standard component
+of existing metabolomics workflows. The availability of high-quality predicted spectra across large chemical
+structure databases would therefore greatly increase compound identification rates in real experimental
+settings.
+2.4
+Mass decomposition
+Modern mass spectrometry achieves sufficiently high resolution to detect small deviations in m/z from
+integrality that are characteristic of different chemical elements. This property is a key strength of the
+technology, because it permits annotating peaks with formulas through mass decomposition [16]. Given a
+product ion of m/z = m, a precursor formula P, and an instrument resolution ϵ, product mass decomposition
+yields a set F(P, m, ϵ) of chemically plausible subformulas of P whose theoretical masses lie within measurement
+error ϵm of m. This can be cast as an integer program, in which all solutions of the following minimization
+with cost ≤ ϵm are enumerated:
+min
+f∈F∗ |⟨µ, f⟩ − m|
+(1)
+s.t. f ≤ P, f ∈ Ω
+(2)
+where Ω describes a general set of constraints that exclude unrealistic molecular formulas [17]. In practice,
+with modern instruments ϵ is sufficiently small for there to typically be only one or a few solutions to this
+optimization. This allows us to later rely on product mass decomposition as a black-box to generate useful
+formula annotations at training time.
+3
+Related work
+Bond-breaking, used in [13, 14, 18], solves the problem of representing the output space by enumerating the
+2D structures of all probable product ions. These are taken to be connected subgraphs of the precursor,
+generated by sequences of edge removals. Each product ion structure is scored for its probability of formation,
+and a spectrum is generated by associating this probability with each structure’s theoretical m/z. Bond-
+breaking therefore achieves perfect m/z resolution, but suffers from two major weaknesses: first, enumerating
+substructures scales poorly with molecule size, and is not conducive to massively-parallel implementation on
+a GPU. We found a state-of-the-art method [13] takes ∼5s on average to predict a single mass spectrum,
+which precludes training on the largest available datasets: using the same settings as its authors, training
+[13] on ∼300k spectra in NIST-20 would take an estimated three months on a 64-core machine. It also poses
+serious limitations at test time, as inference with a large-scale structure database like ChEMBL [19] requires
+predicting millions of spectra. The other weakness of bond-breaking arises from a restrictive modelling
+assumption: rearrangement reactions [20] frequently yield product ions that are not reachable from the
+precursor by sequences of edge removals.1
+Mass-binning is used for spectrum prediction by [10], and subsequently employed in recent preprints [11, 12].
+This approach represents a mass spectrum as a fixed-length vector via discretization: the m/z axis is
+partitioned into narrow regularly-spaced bins, and each bin is assigned the sum of the intensities of all
+1While engineered rules are used in bond-breaking to account for certain well-studied rearrangements, we found the state-of-
+the-art method CFM-ID still fails to assign a formula annotation within 10ppm to 42% of monoisotopic peaks in the NIST-20
+dataset.
+4
+
+peaks falling within its endpoints. Spectrum prediction then becomes a vector-valued regression problem.
+Mass-binning is conducive to GPU implementation and scales better than bond-breaking, but because a target
+space with millions of mass bins is too large, realistic bin counts lose essential high resolution information
+about the molecular formulas of the peaks: discarding a key strength of MS/MS analysis in favor of a tractable
+learning problem. Such models are also susceptible to edge effects, where m/z measurement error of the
+instrument can cause peaks of the same product ion to cross bin boundaries from spectrum to spectrum.
+Other approaches include molecular dynamics simulation [21], which has extremely high computational costs;
+and NLP-inspired models for peptides [22, 23], which are effective but inapplicable to other types of molecules.
+4
+GrAFF-MS
+Our approach constitutes three major components: first, we represent the output space of spectrum pre-
+diction as a space of probability distributions over molecular formulas. We then introduce a constant-sized
+approximation of this output space using a fixed vocabulary of formulas, which we can generate from our
+training data; we later show this introduces only minor approximation cost, as most formulas occur with low
+probability. Finally, we derive a loss function that takes into account data-specific ambiguities introduced by
+our model of the output space. These components together allow us to efficiently predict spectra using a
+standard graph neural network architecture. We call our approach GrAFF-MS: (Gr)aph neural network for
+(A)pproximation via (F)ixed (F)ormulas of (M)ass (S)pectra.
+4.1
+Modelling spectra as probability distributions over molecular subformulas
+of the precursor
+Our aim is to predict a mass spectrum from a molecular graph. To do so, we must determine how to best
+represent the output space: a spectrum comprises a variable-length set of peaks located at continuous m/z
+positions, whose heights sum to one. We notice that peaks are not located arbitrarily: the set of m/zs is
+structured, as the m/z of a peak is determined (up to measurement error) by the molecular formula of its
+corresponding product ion. This formula is sufficient to determine the m/z; in particular, we do not need to
+know the product ion’s full 2D structure. We therefore model a mass spectrum as a probability distribution
+over molecular subformulas F(P) of the precursor P:
+S = {(mf, yf) : f ∈ F(P)}
+(3)
+where mf .= ⟨µ, f⟩ is the theoretical mass of formula f.
+This is more efficient in principle than bond-breaking, which models a spectrum as a distribution over at
+worst exponentially many substructures of the precursor. But the number of subformulas is only polynomial
+in the coefficients of the precursor formula – and the majority of subformulas can be ruled out a priori as
+chemically infeasible [17]. It is also less restrictive than bond-breaking, which relies on hand-engineered
+rules to capture rearrangement reactions: enumerating subformulas is guaranteed to generate all possible
+peaks, irrespective of whether the structure of their product ion is reachable by edge removals or not. Yet
+our approach preserves the core advantage of bond-breaking over mass-binning: predicting a height for each
+subformula yields spectra with perfect m/z resolution.
+4.2
+Fixed-vocabulary approximation of formula space
+In practice, enumeration of subformulas is still a costly operation for larger molecules. One way to avoid
+this would be to sequentially decode formulas of nonzero probability one at a time: we opt not to do so, as
+this requires a more complex, data-hungrier model, and necessitates a linear ordering of formulas, for which
+there is not an obvious correct choice. Instead, we exploit a property of small molecule mass spectra that we
+discovered in this work, and illustrate in Figure 2 of the results: almost all of the signal in small molecule
+mass spectra lies in peaks that can be explained by a relatively small number (∼2%) of product ion and
+neutral loss formulas that frequently recur across spectra.
+5
+
+Inspired by this finding, we approximate F(P) via the union ˆF(P) = ˆP ∪ (P − ˆL) of a fixed set of frequent
+product ion formulas ˆP, and a variable set of ‘precursor-specific’ formulas P − ˆL obtained by subtracting
+a fixed set of frequent neutral loss formulas ˆL from the precursor P. This greatly simplifies the spectrum
+prediction problem: we now only need to predict a probability for each of the formulas in ˆP and ˆL, which we
+can accomplish with time constant in the size of the precursor.
+Stated explicitly, we approximate the spectrum as:
+S ≈ {(mf, yf) : f ∈ ˆF(P)}
+(4)
+where a height of zero is implicitly assigned to any formula not in ˆF(P).
+The fact that we can equally represent a product ion by either its own formula or a neutral loss formula
+relative to its precursor ion is crucial to generalization. If we only included frequent product ion formulas, we
+would explain peaks of low mass well, which typically correspond to small charged functional groups. But as
+formula space becomes larger with increasing mass, it becomes increasingly unlikely that every significant
+peak of higher mass in an unseen compound will be explained. However such peaks do not represent arbitrary
+subformulas of the precursor: they tend to arise from losses of small uncharged functional groups and
+combinations thereof, which we capture by including frequent neutral losses.
+Our algorithm to generate ˆP and ˆL involves listing all product ion and neutral loss formulas yielded by
+mass decomposition of the training set, and ranking them by the sum of the heights of all peaks to which
+each formula is assigned; we select the top K highest ranked among either type. Pseudocode is provided in
+appendix A.
+4.3
+Peak-marginal cross entropy
+To train our approach, we must rely on formula annotations generated by mass decomposition. Because mass
+spectrometers have limited resolution, it is often the case that more than one valid subformula has a mass
+within measurement error of a peak. These are considered equiprobable a priori, and need not be mutually
+exclusive: it is possible for a compound to contain two distinct substructures with m/z difference smaller
+than the measurement error. As we cannot pick a single formula in such cases, we approximate the full cross
+entropy by marginalizing over compatible formulas: this yields the peak-marginal cross entropy, which we
+minimize w.r.t. the parameters of a neural network ˆy(·; θ):
+min
+θ
+−
+N
+�
+n=1
+�
+i∈Sn
+yn
+i log
+�
+f∈ ˆ
+Fn
+i
+ˆyf(Gn; θ)
+(5)
+using the shorthand ˆFn
+i
+.= ˆF(Pn) ∩ F(Pn, mn
+i , ϵ) to indicate the intersection of our approximated vocabulary
+with the formula annotations generated by mass decomposition. We provide a derivation from first principles
+in appendix B.
+In this formulation, given a molecular graph G of a precursor with formula P, our model predicts a probability
+ˆyf for every formula f in the fixed vocabulary. This produces a spectrum ˆS = {(mf, ˆyf) : f ∈ ˆF(P)}. These
+per-formula probabilities are summed within each observed peak across its compatible formulas to yield a
+predicted peak height, and the cross-entropy between the observed and predicted peak heights across the
+entire spectrum is minimized.
+4.4
+Model architecture
+Formulating spectrum prediction as graph classification permits a fairly typical GNN architecture. GrAFF-
+MS uses a graph isomorphism network with edge and graph Laplacian features [24, 25, 26]: this encodes the
+molecular graph into a dense vector representation, which is then conditioned on mass-spectral covariates
+and passed through a feed-forward network that decodes a logit for each formula in the vocabulary.
+We start with the graph of the 2D structure G = (V, E), to which we add a virtual node [27] and four classes
+of features: node features ai ∈ Rdatom, edge features bij ∈ Rdbond, covariate features c ∈ Rdcov, and the top
+6
+
+eigenvectors and eigenvalues of the graph Laplacian vi ∈ Rdeig, λ ∈ Rdeig. We use the canonical atom and
+bond featurizers from DGL-LifeSci [28] to generate a and b. Since a mass spectrum is not fully determined
+by the molecular graph, c includes a number of necessary experimental parameters: normalized collision
+energy, precursor ion type, instrument model, and presence of isotopic peaks. Further details are provided in
+Table 2 of the appendix; there we also provide hyperparameter settings.
+We first embed the node, edge, and covariate features into Rdenc, reusing the following MLP block:
+MLP(·) = LayerNorm(Dropout(SiLU(Linear(·))))
+and transform the Laplacian features into node positional encodings in Rdenc using a SignNet [26] with φ and
+ρ both implemented as 2-layer stacked MLP blocks:
+xatom
+i
+= MLPatom(ai)
+(6)
+xeig
+i
+= SignNet(vi, λ)
+(7)
+xbond
+ij
+= MLPbond(bij)
+(8)
+xcov = MLPcov(c)
+(9)
+taking i ∈ V and (i, j) ∈ E. We sum the embedded atom features and node positional encodings, and pass
+these along with embedded bond features into a stack of alternating L message-passing layers to update the
+node representations, and L MLP layers to update the edge representations.
+x(0)
+i
+= xatom
+i
++ xeig
+i
+(10)
+e(0)
+ij = xbond
+ij
+(11)
+X(l+1) = X(l) + GINEConv(l)(G, X(l), E(l))
+(12)
+e(l+1)
+ij
+= e(l)
+ij + MLP(l)
+edge(e(l)
+ij ∥x(l+1)
+i
+∥x(l+1)
+j
+)
+(13)
+where ∥ denotes concatenation. The message-passing layer uses the GINEConv operation implemented in [29]:
+for its internal feed-forward network, we use two stacked MLP blocks with GraphNorm [30] in place of layer
+normalization. We similarly replace layer normalization with GraphNorm in the MLPedge blocks. Both node
+and edge updates use residual connections, which we found greatly accelerate training.
+We generate a dense representation of the molecule by attention pooling over nodes [31], to which we add
+the embedded covariate features. This is decoded by a feed-forward network into a spectrum representation
+xspec ∈ Rddec.
+ai = Softmaxi∈V (Linear(xi))
+(14)
+xmol =
+�
+i∈V
+aix(L)
+i
+(15)
+xspec = MLPspec(xmol + xcov)
+(16)
+where MLPspec is a stack of L′ MLP blocks with residual connections. In principle we may now project this
+representation via a linear layer (wk, bk) into a logit zk for each of the K product ion or neutral loss formulas
+in the vocabulary.
+4.5
+Domain-specific modifications
+We must now introduce a number of corrections motivated by domain knowledge to produce realistic mass
+spectra.
+Following fragmentation, certain product ions can bind ambient water or nitrogen molecules as adducts,
+shifting the mass of the fragment. This occurrence is annotated in our training set. For each formula in our
+vocabulary, we therefore predict a logit for three different adduct states of the fragment, indexed by α: the
+original product ion f, f + H2O, and f + N2. In effect this triples our vocabulary size.
+7
+
+Depending upon instrument parameters, tandem mass spectra can display small peaks arising from higher
+isotopic states of the precursor ion, at integral m/z shifts relative to the monoisotopic peak. Isotopic state is
+independent of fragmentation: so rather than expanding our vocabulary again, we apply to all predictions a
+shared offset for each isotopic state β ∈ {0, 1, 2}, which we parameterize on xspec.
+As our vocabulary includes both product ions and neutral losses, we must deal with occasional double-counting:
+depending on the precursor P, there are cases where the same subformula f will be predicted both as a
+product ion (f ∈ ˆP) and a neutral loss (P − f ∈ ˆL). In such cases – denoted by the set ˆD(P) – we subtract a
+log 2 correction factor from both logits: this way the innermost summation in Equation 5 takes the mean of
+their contributions instead of their sum.
+Applying these corrections and softmaxing yields the final heights of the predicted mass spectrum ˆy:
+zαβ
+k
+= wα
+k xspec + wβxspec + bα
+k
+(17)
+− I[k ∈ ˆD(P)] log 2
+(18)
+ˆyαβ
+k
+= Softmaxk,α,β(zαβ
+k )
+(19)
+5
+Experiments
+5.1
+Datasets
+5.1.1
+NIST-20
+We train our model on the NIST-20 tandem MS spectral library [32]. This is the largest dataset of high
+resolution mass spectra of small molecules, curated by expert chemists, and is commercially available for a
+modest fee.2 For each measured compound, NIST-20 provides typically several spectra acquired across a
+range of collision energies. Each spectrum is represented as a list of (m/z, intensity, annotation) peak tuples,
+in addition to metadata describing instrumental parameters and compound identity. The annotation field
+includes a list of formula hypotheses per peak that were computed via mass decomposition.
+We restrict NIST-20 to HCD spectra with [M + H]+ or [M − H]− precursor ions. We exclude structures
+that are annotated as glycans or peptides or exceed 1000Da in mass (as these are not typically considered
+small molecules), or have atoms other than {C, H, N, O, P, S, F, Cl, Br, I}.
+We use an 85/5/10 structure-disjoint train/validation/test split, which we generate by grouping spectra
+according to the connectivity substring of their InChiKey [34] and assigning spectra to splits an entire group
+at a time. As CFM-ID only predicts monoisotopic spectra at qualitative energy levels {low, medium, high},
+we restrict the test set to spectra with corresponding energies {20, 35, 50} in which no peaks were annotated
+as higher isotopes. This yields 306,135 (19,871) training, 17,987 (1,167) validation, and 4,548 (1,637) test
+spectra (structures).
+5.1.2
+CASMI-16
+It is well known that uniform train-test splitting can overestimate generalization in molecular machine learning
+[35]. To address this issue, we employ an independent test set: the spectra of the 2016 CASMI challenge
+[36]. This is a small public-domain mass spectrometry dataset, constructed by domain experts specifically
+for testing algorithms, and comprises structures selected as representative of those encountered ‘in the wild’
+when performing mass spectrometry of small molecules.
+We use [M + H]+ and [M − H]− spectra from the combined ‘Training’ and ‘Challenge’ splits from Categories
+2 and 3 of the challenge. We exclude any structures from CASMI-16 with an InChiKey connectivity match
+to any in NIST-20, yielding 156 spectra of 141 structures. The collision energy stepping protocol used in
+2Open data is not the norm in small molecule mass spectrometry, as large-scale annotation requires substantial time
+commitment from teams of highly-trained human experts. As a result, no public-domain dataset exists of comparable scale and
+quality to NIST-20. However, NIST-20 and its predecessors are well-established in academic mass spectrometry, and have
+been used in previous machine learning publications [10, 33].
+8
+
+CASMI-16 is simulated by predicting a spectrum at each of {20, 35, 50} normalized collision energy and
+returning their mean.
+5.2
+Baselines
+5.2.1
+CFM-ID
+CFM-ID [13] is a bond-breaking method, viewed by the mass spectrometry community as the state-of-the-art
+in spectrum prediction [9]. We found CFM-ID prohibitively expensive to train on NIST-20 (one parallelized
+EM iterate on a subset of ∼60k spectra took 10 hours on a 64-core machine) so we use trained weights
+provided by its authors, learned from 18,282 spectra in the commercial METLIN dataset [37]. Domain
+experts consider spectra acquired under METLIN’s conditions interchangeable with those of NIST-20 [38]
+so it is reasonable to evaluate their model on our data.
+5.2.2
+NEIMS
+NEIMS [10] is a feed-forward network that inputs a precomputed molecular fingerprint and outputs a
+mass-binned spectrum, which is postprocessed using a domain-specific gating operation. We retrained
+NEIMS on NIST-20, which necessitated two modifications: (1) we concatenate a vector of covariates to
+the fingerprint vector, without which NIST-20 spectra are not fully determined; and (2) we bin at 0.1Da
+intervals instead of 1Da intervals, to account for finer instrument resolution in NIST-20. We otherwise use
+the same hyperparameter settings as the original paper, and early-stop on validation loss.
+5.3
+Evaluation metrics
+We quantify predictive performance using mass-spectral cosine similarity, which compares spectra subject
+to m/z measurement error [39] by matching pairs of peaks. For two spectra S and ˆS, mass-spectral cosine
+similarity CS, ˆS is the value of the following linear sum assignment problem:
+CS, ˆS
+.=
+max
+xij∈{0,1}
+�
+i∈S, j∈ ˆS:
+|mi− ˆmj|≤τ
+xij
+yi
+∥y∥2
+ˆyj
+∥ˆy∥2
+(20)
+s.t. �
+i∈S xij ≤ 1
+(21)
+�
+j∈ ˆS xij ≤ 1
+(22)
+We use the CosineHungarian implementation in the matchms Python package, with tolerance τ = 0.05.
+We report mean cosine similarity across spectra, as well as the fraction of spectra scoring > 0.7, which is
+commonly employed in spectral library search as a heuristic cutoff for a positive match [40].
+We also compare runtime of GrAFF-MS against the bond-breaking method CFM-ID. For fair comparison,
+we time a forward pass for each structure in the NIST-20 test split using only the CPU, without any batching.
+We include time spent in preprocessing: our input is a SMILES string and experimental covariates, and our
+output is a spectrum. As collision energy affects the number of peaks that CFM-ID generates, we predict
+spectra at low, medium, and high energies and use the average runtime of the three.
+6
+Results
+6.1
+A fixed vocabulary of product ions and neutral losses closely approximates
+most mass spectra
+Figure 2 demonstrates the fraction of ion counts explained in the average mass spectrum as the training
+vocabulary size is varied. This shows most signal lies within peaks explainable by a relatively small number
+of product ion and neutral loss formulas. In particular, the vocabulary we use (of size K = 104) is sufficient
+to explain 98% of ion counts in the NIST-20 training split, which comprises 193,577 unique product ion
+formulas and 348,692 unique neutral loss formulas. We observe that a fixed vocabulary generalizes beyond
+9
+
+Figure 2: Generalization of different heuristics for fixed-size vocabulary selection. For a given vocabulary size
+on the x-axis, the y-axis indicates the sum of all explained peaks’ heights within a given spectrum, averaged
+over all spectra.
+NIST-20 to a separate dataset CASMI-16, indicating ‘formula sparsity’ is a general property of small
+molecule mass spectra and not limited to our particular training set. We also compare alternative strategies
+of picking only the top products or top losses: using both types of formulas explains more signal for a given
+K than either alone.
+6.2
+GrAFF-MS outperforms bond-breaking and mass-binning on standard
+MS/MS datasets
+Table 1 shows GrAFF-MS produces spectra with greater cosine similarity to ground-truth than either
+baseline. More of our spectra also meet the CS ˆS > 0.7 threshold for useful predictions. These results hold
+both for the NIST-20 test split and the independent test set CASMI-16. We see all methods perform better
+on CASMI-16 than NIST-20: this is because NIST-20 includes a minority of substantially larger molecules
+(max weight 986Da) than CASMI-16 (max weight 539Da), with which all three methods struggle.
+Table 1: Mean cosine similarity E[C] and fraction of useful predictions P(C > 0.7) on the NIST-20 test split
+and CASMI-16. 95% confidence intervals are computed via nonparametric bootstrap.
+NIST-20 Test
+(N = 4548)
+CASMI-16
+(N = 156)
+Method
+E[C]
+P(C>0.7)
+E[C]
+P(C>0.7)
+CFM-ID
+.52±.01
+.35±.02
+.75±.05
+.70±.07
+NEIMS
+.60±.01
+.50±.01
+.63±.05
+.54±.08
+GrAFF-MS
+.70±.01
+.62±.02
+.79±.05
+.76±.07
+6.3
+Representing peaks as formulas scales better with molecular weight than
+bond-breaking
+Figure 3 shows our approach to modelling high resolution spectra scales better with input size than bond-
+breaking.
+CFM-ID takes on average 4.9 seconds per structure in the NIST-20 test split, and scales
+quadratically (R2 = 0.78) with input size. (We believe this is because larger molecules in NIST-20 tend to
+be approximately path graphs – e.g. long hydrocarbon chains – with only quadratically many connected
+subgraphs.) In comparison, running GrAFF-MS on the CPU takes 1.3 core-seconds per spectrum, and
+10
+
+1.0
+0.8
+0.6
+Strategy
+0.4
+Products+Losses
+Products
+Losses
+0.2
+Dataset
+NIST-20 (Train)
+NIST-20 (Test)
+0.0
+CASMI-16
+102
+103
+104
+105
+Vocabulary sizeFigure 3: Empirical time complexity on NIST-20 structures with respect to molecular weight. Each dot is
+a structure. Solid lines are quadratic (blue) and linear (red) fits; dotted line indicates an average over all
+spectra computed using shuffled minibatches.
+scales approximately linearly (R2 = 0.65). This pays off at larger molecular weight: for molecules > 500Da,
+our model is 16× faster on average. Realistically, large-scale prediction will use the GPU: on a single GPU
+with batch size 512, predicting all of the NIST-20 test spectra averages to 2.8ms per spectrum (mostly spent
+in preprocessing on the CPU).
+We can additionally estimate the time required for each approach to generate a reference library from all
+2,259,751 structures below 1000Da in ChEMBL v3.1. Correcting for greater mean molecular weight in
+ChEMBL (405Da vs 292Da), this would take 2 hours running our PyTorch research code as-is on a single
+GPU. The same library would take 4 days to generate with 64 parallel instances of CFM-ID, which is written
+in optimized C++ code – showing the importance of efficiently representing the output space.
+6.4
+GrAFF-MS distinguishes very similar compounds and makes human-like
+mistakes
+In Figure 4 we show some particularly challenging examples of mass spectra. The top and middle panels
+show two structurally similar compounds, differing only by the order of a carbon-carbon bond. Our approach
+correctly predicts distinct spectra for each (CS ˆS = 0.98, top; CS ˆS = 0.95, middle). The third molecule is an
+example where we fail to predict a realistic spectrum (CS ˆS = 0.03), but in a manner in which a human expert
+would also fail. This molecule is a member of the phthalate class, which chemists recognize by a characteristic
+dominant peak at 149Da [41]. Our model predicts this same peak, correctly recognizing a phthalate: but in
+this case that peak is relatively minor, indicating atypical fragmentation chemistry.
+7
+Discussion
+In this work we develop GrAFF-MS, a graph neural network for predicting high resolution mass spectra of
+small molecules. Unlike previous approaches that force a tradeoff between m/z resolution and a tractable
+learning problem, GrAFF-MS is both computationally efficient and capable of modelling the high-resolution
+m/z information essential to mass spectrometry. This is made possible by our discovery that mass spectra of
+small molecules can be closely approximated as distributions over a fixed vocabulary of molecular formulas,
+highlighting the value that domain-aware modelling can add to molecular machine learning. Particularly
+surprising was that we outperform CFM-ID, which trades model expressivity for an even stronger scientific
+prior that we expected would contribute to better generalization. However, this prior incurs a heavy cost in
+11
+
+102
+101
+100
+Time
+10-1
+CFM-ID
+GrAFF-MS (CPU)
+GrAEE-MS(GPU)
+10-2
+200
+400
+600
+800
+1000
+Molecular weight (Da)Figure 4: Three compounds from CASMI-16, with spectra predicted by our model (blue) against negated
+ground-truth (orange). Oxygens are shaded red by convention.
+time complexity, making it impractical to train CFM-ID on hundreds of thousands of spectra as we did.
+Future directions include pretraining on large-scale property prediction tasks, revisiting sequential formula
+decoding, and incorporating additional scientific priors about fragmentation chemistry. Overall we anticipate
+this work will both accelerate scientific discovery and demonstrate mass spectrometry to be a compelling
+domain for continued machine learning research.
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+radical cation complexes with neutral molecules. Journal of the American Society for Mass Spectrometry,
+22(11), August 2011.
+[42] Diederik P Kingma and Jimmy Ba. Adam: A method for stochastic optimization. 2015.
+15
+
+A
+Fixed vocabulary selection
+Algorithm 1 describes our procedure for selecting the product ions ˆP and neutral losses ˆL. We use the
+shorthand Fn
+i = F(Pn, mi, ϵ) to indicate the set of formulas computed by mass decomposition. When mass
+decomposition yields more than one formula annotation for a peak, here we split the peak height uniformly
+among all annotations.
+Algorithm 1 Fixed vocabulary selection
+Input: training spectra and precursors {(Sn, Pn)}N
+n=1, vocabulary size K, tolerance ϵ
+Initialize Ff = 0, Ll = 0, ˆP = ∅, ˆL = ∅
+for n = 1 . . . N do
+for (mi, yi) ∈ Sn do
+for f ∈ Fn
+i do
+l = Pn − f
+Ff = Ff + yi/|Fn
+i |
+Ll = Ll + yi/|Fn
+i |
+end for
+end for
+end for
+Sort F and L
+while | ˆP| + | ˆL| ≤ K do
+f = first element of F
+l = first element of L
+if Ff > Ll then
+Add f to ˆP
+Remove f from F
+else
+Add l to ˆL
+Remove l from L
+end if
+end while
+B
+Derivation of peak-marginal cross entropy
+We derive our loss function from physical first principles, making a number of minor modelling assumptions:
+• The number of precursor ions accumulated by the instrument is Poisson with rate λ.
+• Each individual precursor ion is independently converted into fragment f with probability pf.
+• The instrument resolution parameter ϵ is sufficiently small that separate peaks do not overlap: there
+exists exactly one peak i(f) for every f : pf > 0 satisfying |⟨µ, f⟩ − mi| ≤ ϵmi.
+By the splitting property, the number of ions of each fragment are independently Poisson with rate λf = λpf.
+By the merging property, the height of peak i is also a Poisson r.v. Ki with rate λi = �
+f∈Fi λf, where Fi
+denotes the set of fragments whose theoretical masses all fall within the measurement error ϵmi of peak i.
+16
+
+The log-likelihood for the peak height is (taking equality up to constants C w.r.t. pf):
+log P(Ki = ki)
+(23)
+= ki log λi − λi − log ki!
+(24)
+= ki log
+�
+� �
+f∈Fi
+λf
+�
+� −
+�
+� �
+f∈Fi
+λf
+�
+� + C
+(25)
+= ki log
+�
+� �
+f∈Fi
+λpf
+�
+� −
+�
+� �
+f∈Fi
+λpf
+�
+�
+(26)
+= ki log λ + ki log
+�
+� �
+f∈Fi
+pf
+�
+� − λ
+�
+f∈Fi
+pf
+(27)
+= C + ki log
+�
+� �
+f∈Fi
+pf
+�
+� − λ
+�
+f∈Fi
+pf
+(28)
+where (25) uses merging, and (26) uses splitting. Because each fragment is assigned to exactly one peak
+(no overlap), the peak heights {Ki : i ∈ S} are independent. Let the total number of accumulated ions
+K = �
+i∈S ki in spectrum S. Defining yi = ki/K:
+log P({Ki = ki : i ∈ S})
+(29)
+=
+�
+i∈S
+log P(Ki = ki)
+(30)
+=
+�
+i∈S
+�
+�ki log
+�
+� �
+f∈Fi
+pf
+�
+� − λ
+�
+f∈Fi
+pf
+�
+�
+(31)
+=
+�
+i∈S
+(Kyi) log
+�
+� �
+f∈Fi
+pf
+�
+� − λ
+�
+i∈S
+�
+f∈Fi
+pf
+(32)
+= K
+�
+i∈S
+yi log
+�
+� �
+f∈Fi
+pf
+�
+� − λ · 1
+(33)
+= C
+�
+i∈S
+yi log
+�
+� �
+f∈Fi
+pf
+�
+� + C′
+(34)
+where (33) again uses our assumption that every fragment is assigned to exactly one peak. Dropping the
+constants and negating the final term yields the peak-marginal cross-entropy loss for a single spectrum.
+17
+
+C
+Model hyperparameters
+We use a vocabulary of K = 10000 formulas. We train an L = 6-layer encoder and L′ = 2-layer decoder
+with denc = 512 and ddec = 1024, resulting in 44.6 million trainable parameters. We use the deig = 8
+lowest-frequency eigenvalues, truncating or padding with zeros. All dropout is applied at rate 0.1. We use a
+batch size of 512 and the Adam optimizer [42] with learning rate 5 × 10−4. We train for 100 epochs and use
+the model from the epoch with the lowest validation loss. All models are trained using PyTorch Lightning
+with automatic mixed precision on 2 Tesla V100 GPUs.
+D
+Mass spectral covariates
+Table 2: Mass spectral covariates used in our model.
+Feature
+Range
+Comment
+Normalized collision energy
+[0, 200]
+Thermo Scientific PSB104,
+“Normalized Collision Energy Technology”
+Precursor type
+[M + H]+, [M − H]−
+Includes ionization mode & adduct composition
+Instrument model
+Orbitrap Fusion Lumos,
+Thermo Finnigan Elite Orbitrap,
+Thermo Finnigan Velos Orbitrap
+Different limits of detection
+Has isotopic peaks
+False, True
+Proxy for width setting of precursor mass filter
+18
+

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@@ -0,0 +1,1316 @@
+arXiv:2301.05546v1  [physics.chem-ph]  13 Jan 2023
+Comparing real-time coupled cluster methods through simulation of collective Rabi
+oscillations
+Andreas S. Skeidsvoll
+Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway
+Henrik Koch∗
+Scuola Normale Superiore, Piazza dei Cavalieri, 7, I-56126, Pisa, Italy
+and
+Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway
+(Dated: January 16, 2023)
+The time-dependent equation-of-motion coupled cluster (TD-EOM-CC) and time-dependent cou-
+pled cluster (TDCC) methods are compared by simulating Rabi oscillations for different numbers
+of non-interacting atoms in a classical electromagnetic field. While the TD-EOM-CC simulations
+are numerically stable, the oscillating time-dependent energy scales unreasonably with the num-
+ber of subsystems resonant with the field. The TDCC simulations give the correct scaling of the
+time-dependent energy in the initial stages of the Rabi cycle, but the numerical solution breaks
+down when the multi-atom system approaches complete population inversion. We present a general
+theoretical framework in which the two methods can be described, where the cluster amplitude time
+derivatives are taken as auxiliary conditions, leading to a shifted time-dependent Hamiltonian ma-
+trix. In this framework, TDCC has a shifted Hamiltonian with a block upper triangular structure,
+explaining the correct scaling properties of the method.
+I.
+INTRODUCTION
+With recent developments in the shaping and am-
+plification of laser pulses, the production of short and
+strong pulses can now be realized in several frequency
+domains [1–3]. The progress sparks further interest in
+the dynamical and non-linear response of molecules to
+strong fields, which can involve a high number of quan-
+tum states [4]. This since many of the states that are
+inaccessible by a single-photon transition, either energet-
+ically or by symmetry selection rules, can be accessed by
+a multiphoton transition [5, 6].
+The ultrafast non-linear response of molecules to
+strong fields can give an extended degree of dynamic con-
+trol of chemical reactions [7–9]. It can also reveal infor-
+mation about the system that is inaccessible in weaker
+fields, which can be used for improving the imaging of dif-
+ferent reaction stages [5, 8]. That said, the involved cou-
+pling between the numerous affected states can lead to
+an intricate relationship between the shape and strength
+of laser pulses and the molecular response, which calls
+for the interpretation by appropriate quantum chemistry
+methods [10].
+Both the accuracy and computational complexity of
+quantum chemistry methods often increase with the or-
+der of approximation of the particle correlations in the
+system [10, 11] and the size of the finite basis set [12], but
+the accuracy also depends heavily on the mathematical
+structure of the method. Considerable effort is spent on
+constructing the most well-behaved methods for a given
+order of computational complexity, with respect to both
+numerical stability and correspondence to experimental
+∗ [email protected]
+results for various systems. Real-time variants of quan-
+tum chemistry methods are convenient for modeling mul-
+tiphoton transitions in systems [13], as expressions for the
+high-order frequency response can be difficult to both de-
+rive and to solve numerically.
+The well-established single-reference coupled cluster
+(CC) hierarchy of methods often gives accurate and
+rapidly converging molecular properties [14] for states
+with weak multi-reference character [15].
+An impor-
+tant reason for this accuracy is the physically reason-
+able scaling properties of the methods, even when the
+cluster operator is truncated. For instance, the energy
+of the ground state is size-extensive, meaning that it
+scales linearly with the number of non-interacting iden-
+tical subsystems [16]. In the equation-of-motion frame-
+work, the excitation energies are size-intensive, mean-
+ing that they do not scale with the number of non-
+interacting subsystems [17]. In the linear response frame-
+work, which is based on time-dependent coupled cluster
+theory, ground state–excited state transition moments
+are also size-intensive [18]. Truncated configuration in-
+teraction methods, on the other hand, do not possess
+these properties, and errors generally increase with the
+size of the simulated system [19].
+Traditionally, the coupled cluster methods have almost
+exclusively been treated in the frequency domain, but
+the last decade has witnessed an increased exploration
+into their real-time behavior [13]. As demonstrated by
+Pedersen and Kvaal [20] and further investigated by
+Kristiansen et al.
+[21], the exponential parametriza-
+tion makes the standard time-dependent coupled cluster
+(TDCC) method inherently unstable whenever the refer-
+ence determinant weight is depleted by a strong field.
+These instabilities can require the use of exceedingly
+small time steps in numerical solutions, and can at times
+also lead to breakdowns that cannot be solved by de-
+
+2
+creasing the time step size.
+The orbital adaptive time-dependent coupled clus-
+ter (OATDCC) method, which requires the solution
+of an additional set of linear equations at each time
+step, was shown to have a greater stability region than
+TDCC. Nonetheless, the method still fails at higher field
+strengths, where the reference determinant weight can
+become greater than one [21].
+Variants of the time-dependent equation-of-motion
+coupled cluster (TD-EOM-CC) method have also been
+used for modeling laser-molecule interactions, but only
+a handful applications have included the full non-linear
+real-time propagation of the laser-driven electron dy-
+namics [13, 22–24].
+In these cases, the TD-EOM-CC
+equations were expressed in the basis obtained by diag-
+onalizing the field-free equation-of-motion coupled clus-
+ter Hamiltonian.
+We instead express the equations in
+the elementary basis, leading to equations that are sim-
+ple to implement and have computational and memory
+requirements that scale more favorably with respect to
+system size than the full diagonal basis equations. This
+makes the formulation particularly useful for assessing
+the short-time and non-linear behavior of the TD-EOM-
+CC method.
+The paper is organized as follows. In Section II, the
+TDCC and TD-EOM-CC methods are described in a gen-
+eral framework, and it is shown how the time derivative
+of the cluster amplitudes affects the analytical scaling
+properties of the two methods. Section III outlines the
+computational methods used to simulate atoms undergo-
+ing semiclassical Rabi oscillations in a resonant electro-
+magnetic field. In Section IV, results of the simulations
+are presented and discussed, including a demonstration
+of how the time-dependent energy scales with respect to
+system size in the two methods. The key findings are
+summarized in Section V.
+II.
+THEORY
+A.
+System
+The time-dependent system of the molecule and the
+external field is described by the Hamiltonian
+H(t) = H(0) + V (t).
+(1)
+The field-free molecular system is described by the
+Hamiltonian H(0), and the interaction between the
+molecular system and the external field is described by
+V (t).
+We describe the interaction semi-classically, in
+the dipole approximation and length gauge. This gives
+V (t) = −µ · E(t), where µ is the electric dipole moment
+vector and E(t) the time-dependent electric field vector.
+The system is also treated within the Born-Oppenheimer
+approximation, with fixed nuclei.
+B.
+Time dependence in coupled cluster methods
+Coupled cluster ket and bra vectors that encompass
+both the TDCC and TD-EOM-CC parametrizations can
+be defined as
+|Ψ(t)⟩ = eT (t)R(t) |HF⟩ ,
+(2)
+⟨�Ψ(t)| = ⟨HF| L(t)e−T (t),
+(3)
+where the cluster operator
+T (t) =
+�
+κ≥0
+τκtκ(t)
+(4)
+and the right and left EOM-CC operators
+R(t) =
+�
+κ≥0
+τκrκ(t),
+L(t) =
+�
+κ≥0
+lκ(t)�τ†
+κ.
+(5)
+The operators with index 0 are the unit operator
+τ0 = �τ †
+0 =
+1,
+(6)
+and the operators τµ and �τ †
+µ, where µ > 0, excite and
+deexcite electrons between occupied and virtual Hartree-
+Fock molecular orbitals, respectively,
+τµ |HF⟩ = |µ⟩ ,
+⟨HF| �τ†
+µ = ⟨�µ| ,
+(7)
+�τ †
+µ |HF⟩ = 0,
+⟨HF| τµ = 0.
+(8)
+The operators are chosen so that the excited determi-
+nants are biorthonormal,
+⟨�κ|λ⟩ = δκλ,
+κ ≥ 0,
+λ ≥ 0,
+(9)
+where δκλ is the Kronecker delta. The field-free coupled
+cluster ground state can be defined by setting the am-
+plitudes r(0)
+µ
+= 0 and r(0)
+0
+= l(0)
+0
+= 1, and letting the
+ground state cluster amplitudes t(0)
+µ
+and left vector l(0)
+µ
+be determined as solutions of the field-free ground state
+equations
+⟨�µ| e−T (0)H(0)eT (0) |HF⟩ = 0,
+(10)
+�
+⟨HF| +
+�
+µ>0
+l(0)
+µ ⟨�µ|
+�
+[e−T (0)H(0)eT (0), τν] |HF⟩ = 0.
+(11)
+For simplicity, we also set the undetermined phase-
+related cluster amplitude t(0)
+0
+= 0.
+The equations for the time dependence of the param-
+eters of Eq. (5) can be derived from the right and left
+time-dependent Schr¨odinger equations (TDSEs)
+i d
+dt |Ψ(t)⟩ = H(t) |Ψ(t)⟩ ,
+(12)
+−i d
+dt ⟨�Ψ(t)| = ⟨�Ψ(t)| H(t).
+(13)
+
+3
+Inserting Eq. (2) into Eq. (12) before projecting onto
+⟨�κ| e−T (t), and likewise inserting Eq. (3) into Eq. (13) be-
+fore projecting onto eT (t) |λ⟩, the following matrix-vector
+TDSEs are obtained
+idrκ(t)
+dt
+=
+�
+λ≥0
+�Hκλ(t)rλ(t),
+(14)
+−idlλ(t)
+dt
+=
+�
+κ≥0
+lκ(t) �Hκλ(t),
+(15)
+where the shifted Hamiltonian
+�H(t) = H(t) − idT (t)
+dt
+.
+(16)
+The elements of the coupled cluster matrix O(t) of oper-
+ator O(t) are given by
+Oκλ(t) = ⟨�κ| O(t) |λ⟩ ,
+(17)
+where an overbar is used to denote the similarity trans-
+formation by the exponentiated time-dependent cluster
+operator,
+O(t) = e−T (t)O(t)eT (t).
+(18)
+In TD-EOM-CC, the time derivatives of the cluster
+amplitudes are given by
+idtκ(t)
+dt
+= 0,
+(19)
+while in TDCC, the derivatives are given by
+idtκ(t)
+dt
+= ⟨�κ| H(t) |HF⟩ .
+(20)
+The resolution of identity
+1 = �
+η≥0 |η⟩ ⟨�η| and Eq. (16)
+can be used to rewrite the matrix elements of the shifted
+Hamiltonian in TDCC as
+�Hκλ(t) = ⟨�κ| H(t) |λ⟩ −
+�
+η≥0
+⟨�κ| τη |λ⟩ ⟨�η| H(t) |HF⟩
+= ⟨�κ|
+�
+H(t), τλ
+�
+|HF⟩ .
+(21)
+Once all time-dependent amplitudes have been found
+at a given point in time t, the time-dependent expecta-
+tion values of the time-dependent operator O(t) can be
+obtained by
+⟨O(t)⟩ = ⟨�Ψ(t)| O(t) |Ψ(t)⟩
+=
+�
+κ,λ≥0
+lκ(t)Oκλ(t)rλ(t)
+= lT (t)O(t)r(t).
+(22)
+C.
+Scaling properties of real-time coupled cluster
+methods
+In order to theoretically investigate the scaling proper-
+ties of methods based on the parametrization in Eq. (2)
+and Eq. (3), we assume that the system is composed of
+non-interacting subsystems.
+We let τλI denote an ele-
+mentary excitation operator and �τ †
+κI an elementary deex-
+citation operator of subsystem I. The elementary excita-
+tion and deexcitation operators of the composite system
+can be constructed as tensor products of all the oper-
+ators of the different subsystems. Untruncated TDCC
+and TD-EOM-CC methods can represent all tensor prod-
+ucts, since the excitation and deexcitation levels of the
+methods are not limited. In truncated methods, however,
+all elementary excitation and deexcitation operators that
+exceed a truncation level specific to the method are ex-
+cluded, which can lead to errors related to the scaling
+from one to several subsystems.
+For two subsystems I ∈ {A, B}, the elementary exci-
+tation and deexcitation operators of the composite sys-
+tem can be constructed as the tensor products τλA ⊗ τλB
+and �τ †
+κA ⊗ �τ †
+κB. We split the sets of these operators into
+four partitions, which we label by 0, A, B and AB. The
+0 partition includes the operators that do not change
+the excitation level of the subsystems, τ0A ⊗ τ0B and
+�τ†
+0A ⊗ �τ †
+0B. The A partition includes the operators that
+change the excitation level of subsystem A only, τµA ⊗τ0B
+and �τ †
+µA ⊗ �τ †
+0B, and the B partition the operators that
+change the excitation level of subsystem B only, τ0A ⊗τµB
+and τ †
+0A ⊗ τ †
+µB, where µ > 0. The AB partition includes
+the operators that change the excitation level of both
+subsystems, τνA ⊗ τνB and �τ†
+µA ⊗ �τ†
+µB, where µ > 0 and
+ν > 0. Truncation can affect the AB partition, since the
+tensor products of the truncated subsystem operators can
+include excitations and deexcitations that in combination
+go beyond the truncation level of the method. In the fol-
+lowing, we assess the effect this truncation has for the
+TDCC and TD-EOM-CC methods.
+We start by assuming that the cluster amplitudes cor-
+responding to the operators τνA ⊗ τνB are zero at a given
+time t. The cluster operator T (t) can then be written as
+the tensor sum,
+T (t) = TA(t) ⊗ IB + IA ⊗ TB(t),
+(23)
+where TI(t) is the cluster operator for subsystems I. Since
+operators on non-interacting subsystems commute, we
+have that
+e±
+�
+TA(t)⊗IB+IA⊗TB(t)
+�
+= e±TA(t) ⊗ e±TB(t).
+(24)
+We furthermore let O(t) be any operator that operates
+independently on the two subsystems and thus can be
+written as the tensor sum
+O(t) = OA(t) ⊗ IB + IA ⊗ OB(t),
+(25)
+where OA(t) and OB(t) are subsystem operators. Equa-
+tion (24) then implies that the similarity transformed
+operator in Eq. (18) can be written as the tensor sum
+O(t) = e−TA(t)OA(t)eTA(t) ⊗ IB
++ IA ⊗ e−TB(t)OB(t)eTB(t)
+= OA(t) ⊗ IB + IA ⊗ OB(t).
+,
+(26)
+
+4
+which does not contain terms where both two subsystems
+are excited simultaneously.
+We furthermore assume that the time-dependent
+Hamiltonian H(t) can be written on the form of Eq. (25).
+In TDCC, the time derivative of the cluster amplitudes
+in Eq. (20) can for the AB partition be written as
+idtµAµB(t)
+dt
+=
+�
+⟨�µA| ⊗ ⟨�µB|
+�
+×
+�
+HA(t) ⊗ IB + IA ⊗ HB(t)
+�
+×
+�
+|HFA⟩ ⊗ |HFB⟩
+�
+= 0.
+(27)
+As long as Eq. (23) holds at the initial time, it will thus
+in TDCC also hold for later times. This is also trivially
+the case for TD-EOM-CC, since the time derivative given
+by Eq. (19) is zero for all cluster amplitudes.
+We let the subscript ∥ denote vectors where the el-
+ements that truncated methods fail to represent have
+been set to zero. Accordingly, the right and left trans-
+pose vectors r∥ and lT
+∥ are the truncated counterparts of
+the untruncated vectors r and lT , and have the following
+representation in the partitioned bases
+r∥ =
+
+
+
+r0
+rA
+rB
+(rAB)∥
+
+
+ ,
+lT
+∥ =
+�l0 lA lB (lAB)∥
+�
+,
+(28)
+where only the AB partitions can be affected by the trun-
+cation. Furthermore, O∥ is the truncated counterpart of
+the operator matrix O, which is the projection of Eq. (26)
+onto the tensor product bases. In the partitioned bases,
+the matrix has the representation
+O∥ =
+
+
+
+O0 0
+O0 A
+O0 B
+0
+OA 0
+OA A
+0
+(OA AB)∥
+OB 0
+0
+OB B
+(OB AB)∥
+0
+(OAB A)∥ (OAB B)∥ (OAB AB)∥
+
+
+ . (29)
+The block matrix maps partitions of right and left trans-
+pose vectors to partitions where the numbers of excited
+subsystems have changed by at most one. Thus, a single
+matrix transformation is not enough to map between the
+0 and AB partitions, and the expectation value involv-
+ing the ground state right vector r(0)
+∥
+= (1, 0, 0, 0)T is
+unaffected by product basis truncation,
+⟨O⟩∥ = lT
+∥ O∥r(0)
+∥
+= lT
+0 O0 0 + lT
+AOA 0 + lT
+BOB 0
+= ⟨O⟩ .
+(30)
+For second- and higher-order transformations, all parti-
+tions of the transformed right and left transpose vectors
+can be affected by the truncation.
+We now assess the effect of transformation by an oper-
+ator matrix T ∥ with the following block upper triangular
+structure
+T ∥ =
+
+
+
+T0 0 T 0 A
+T 0 B
+0
+0
+T A A
+0
+(T A AB)∥
+0
+0
+T B B
+(T B AB)∥
+0
+0
+0
+(T AB AB)∥
+
+
+ .
+(31)
+The block matrix T maps partitions of right vectors to
+themselves and to partitions where the numbers of ex-
+cited subsystems have decreased by one. Thus, the 0 par-
+tition of the ground state right vector r(0) is only mapped
+to itself under successive transformations by block upper
+triangular matrices,
+T ′···′
+∥
+· · · T ′
+∥T ∥r(0)
+∥
+=
+
+
+
+
+(T ′···′ · · · T ′T r(0))0
+0
+0
+0
+
+
+
+ ,
+(32)
+where all truncated matrices T ∥ have been replaced by T
+whenever the truncated AB partition does not make any
+contribution. We can see that the truncation does not
+affect the transformed vector in Eq. (32) at all, while it
+affects all partitions of right vectors transformed by two
+or more matrices with the structure of O. Furthermore,
+T maps partitions of left transpose vectors to themselves
+and to partitions where the numbers of excited subsys-
+tems have increased by one. Thus, the AB partition of
+left transpose vectors is only mapped to itself under suc-
+cessive transformations by block upper triangular matri-
+ces, and
+lT
+∥ T ∥T ′
+∥ · · · T ′···′
+∥
+=
+
+
+
+
+(lT T T ′ · · · T ′···′)0
+(lT T T ′ · · · T ′···′)A
+(lT T T ′ · · · T ′···′)B
+(lT
+∥ T ∥T ′
+∥ · · · T ′···′
+∥
+)AB
+
+
+
+
+T
+,
+(33)
+We can see that the truncation only affects the AB parti-
+tions of the transformed left transpose vector in Eq. (33),
+while it affects all partitions of left transpose vectors
+transformed by two or more matrices with the structure
+of O.
+In the truncated product basis, the exact solutions of
+the right and left matrix TDSEs in Eq. (14) and Eq. (15)
+can be given by
+r∥(t) = U ∥(t, t0)r∥(t0),
+lT
+∥ (t) = lT
+∥ (t0)U ∥(t0, t), (34)
+where
+U ∥(t, t0) = 1∥ − i
+� t
+t0
+dt′ �
+H∥(t′)
++ (−i)2
+� t
+t0
+dt′
+� t′
+t0
+dt′′ �
+H∥(t′)�
+H∥(t′′) + · · · .
+(35)
+Under the conditions that the matrix �
+H(t) has the block
+upper triangular structure of T in Eq. (31), and that
+
+5
+the right vector starts out as the ground state vector
+r(t0) = r(0), Eq. (32) implies that
+r(t) = r∥(t) =
+
+
+
+
+�
+U(t, t0)r(t0)
+�
+0
+0
+0
+0
+
+
+
+ ,
+(36)
+where U(t, t0) denotes the time evolution operator with
+the untruncated time-dependent Hamiltonian matrix
+�
+H(t). Furthermore, Eq. (33) implies that
+l∥(t) =
+
+
+
+
+
+�
+lT (t0)U(t0, t)
+�
+0
+�
+lT (t0)U(t0, t)
+�
+A
+�
+lT (t0)U(t0, t)
+�
+B
+�
+lT
+∥ (t0)U ∥(t0, t)
+�
+AB
+
+
+
+
+ .
+(37)
+Under these conditions, time-dependent expectation val-
+ues are not affected by the truncation of the product basis
+⟨O(t)⟩∥ = lT
+∥ (t)O∥(t)r∥(t)
+=
+�
+lT (t0)U(t0, t)
+�
+0O0 0(t)
+�
+U(t, t0)r(t0)
+�
+0
++
+�
+lT (t0)U(t0, t)
+�
+AOA 0(t)
+�
+U(t, t0)r(t0)
+�
+0
++
+�
+lT (t0)U(t0, t)
+�
+BOB 0(t)
+�
+U(t, t0)r(t0)
+�
+0
+= ⟨O(t)⟩ ,
+(38)
+and thus behave correctly when the system scales from
+one to two non-interacting subsystems. Furthermore, the
+expectation values can be shown to scale correctly to
+any number of non-interacting subsystems by repeatedly
+splitting one of the remaining composite subsystems in
+two before repeating the above arguments.
+From Eq. (21), we can see that the blocks �
+HA 0(t)
+�
+HB 0(t), �
+HAB A(t) and �
+HAB B(t) below the diagonal of
+the shifted Hamiltonian matrix are equal to zero in trun-
+cated TDCC methods.
+This implies that the shifted
+time-dependent Hamiltonian �
+H(t) has the same triangu-
+lar block structure as T in Eq. (31), and the method has
+the correct scaling properties when the system starts out
+in the ground state, in accordance with Eq. (38). Note
+that the amplitude derivative i dt0(t)
+dt
+= ⟨HF| H(t) |HF⟩ in
+Eq. (20) also implies that �H00(t) is zero, but this con-
+dition is not needed for the correctness of the scaling
+properties of truncated TDCC methods.
+In general, however, time-dependent expectation val-
+ues do not have to be same in truncated and untruncated
+methods,
+⟨O(t)⟩∥ = lT
+∥ (t)O∥(t)r∥(t) ̸= lT (t)O(t)r(t) = ⟨O(t)⟩ .
+(39)
+As stated in Eq. (19), the time derivatives of the clus-
+ter amplitudes are zero in TD-EOM-CC, and therefore
+�
+H(t) = H(t). The interaction term V (t) of the Hamil-
+tonian is in general not block upper triangular, and can
+map partitions of right and left transpose vectors in the
+same way as O.
+The truncation of the product basis
+can thus affect all partitions of the right and left vectors,
+and time-dependent expectation values of TD-EOM-CC
+are in general misrepresented when the product basis is
+truncated, in accordance with Eq. (39).
+However, the
+field-free term H0 of the time-dependent Hamiltonian
+has the block upper triangular structure of T , and the
+truncation of the product basis does thus not affect ex-
+pectation values when there is no interaction with the
+external field.
+III.
+COMPUTATIONAL DETAILS
+In order to numerically assess the behavior of TD-
+EOM-CC, we implement the method described by the
+differential equations Eq. (14), Eq. (15) and Eq. (19) in
+the spin-adapted elementary basis, and the expectation
+value expression Eq. (22) in a development version of
+the eT program [25]. We furthermore use the existing
+spin-adapted ground state and TDCCSD methods in eT
+1.0 [25, 26]. The methods are used to calculate the in-
+teraction of atoms with the electromagnetic field repre-
+sented by the electric field
+E(t) = E0ǫ cos(ω0(t − t0) + φ)f(t)
+(40)
+where ǫ0 is the peak field strength, ǫ the polarization, ω0
+the carrier frequency, φ the carrier-envelope phase and
+f(t) the envelope of the field. The envelope is given the
+functional form
+f(t) =
+
+
+
+
+
+0,
+t < a,
+sin2�
+2π(t−a)
+4(b−a)
+�
+,
+a ≤ t ≤ b,
+1,
+t > b,
+(41)
+which increases from zero to one in the interval from a
+to b.
+The cc-pVTZ basis set is used for the helium and beryl-
+lium atoms in the simulations. The laser field is given
+a carrier frequency of 1.880 433 92a.u., corresponding to
+the transition between the ground 0 1S0 state and the first
+dipole-allowed excited 2 1P1 state of helium. The field is
+furthermore given a peak field strength of 2.5 × 10−2 a.u.,
+a polarization in the z-direction and a carrier-envelope
+phase of φ = −π/2. The envelope of the field is set to
+increase from a = 0 a.u. of time until b = 25 optical cy-
+cles (≈83.534a.u.). The envelope gives the laser field a
+narrow bandwidth, centered around the 0 1S0–2 1P1 res-
+onance, which ensures that the state is kept in a time-
+dependent superposition dominated by the two states.
+The integration of the time-dependent differential equa-
+tions is done using the Dormand-Prince method of order
+5(4) with the adaptive time stepping procedure described
+in Appendix B of Ref. [24]. The initial time step size
+is set to 1 × 10−2 a.u., and the maximum and minimum
+values of the estimated error are set to 1 × 10−7 a.u. and
+1 × 10−9 a.u., respectively.
+
+6
+0
+200
+400
+600
+800
+1000
+t (a.u.)
+0.0
+0.5
+1.0
+1.5
+2.0
+2.5
+∆⟨H(t)⟩ (Ha)
+TDCC
+TD-EOM-CC
+FIG. 1. Time-dependent TDCC and TD-EOM-CC simula-
+tions of a single helium atom in a slowly ramped laser field.
+The time-dependent energy difference ∆ ⟨H(t)⟩ = ⟨H(t)⟩−E0
+of each simulation is shown as a function of time.
+IV.
+RESULTS AND DISCUSSION
+A.
+Simulating single-subsystem Rabi oscillations
+with TDCC and TD-EOM-CC
+For a single helium atom, the TDCCSD and TD-EOM-
+CCSD methods can describe all possible excitations of
+the reference determinant, and the time-dependent ob-
+servables are thus analytically equal for the two meth-
+ods. In Fig. 1, we demonstrate that this is also the case
+numerically for the time-dependent excitation energy, as
+the results are equal for the two methods. The excita-
+tion energy starts out at zero, and periodically increases
+and decreases as a function of time, illustrating that the
+system undergoes Rabi oscillation between the 0 1S0 and
+2 1P1 states. TDCC is known to be numerically unstable
+when the weight of the reference determinant approaches
+zero [20, 21], but we observe that the method can be used
+for simulating an essentially complete population inver-
+sion for the single helium atom.
+B.
+Simulating collective Rabi oscillations with
+TDCC
+To numerically investigate the scaling properties of
+TDCC, we compare the results from the single-helium
+simulation with results from simulations of two effectively
+non-interacting helium atoms, where one is placed at the
+origin and the other at 1000 ˚A on the x-axis, respectively.
+The time-dependent excitation energy of each simulation
+is shown in the top row of Fig. 2. Until around 200 a.u.
+of time, we can see that the excitation energy of the two-
+helium TDCC simulation, shown in the second column,
+is two times the excitation energy of the single-helium
+simulation in the first column.
+This is in accordance
+with the theory in Section II, and implies that TDCCSD
+treats the correlation exactly in this interval, even though
+the system has four electrons.
+We also note that the
+combined single-subsystem norms
+�
+∥tA∥2 + ∥tB∥2 and
+�
+∥lA∥2 + ∥lB∥2 of the two-helium calculations, shown
+in the second and third rows, respectively, are
+√
+2 times
+the respective single-helium norms ∥tA∥ and ∥lA∥ in the
+same interval.
+After around 200 a.u. of time, the two-helium solution
+in the second column of Fig. 2 breaks down, and the val-
+ues of the two-helium excitation energies and norms blow
+up. Note that the norm of the AB partition of the clus-
+ter amplitudes of the regular TDCC calculation, shown
+in the bottom left panel, starts out as very small (less
+than 1 × 10−10), but grows continually during the prop-
+agation. As the solution approaches the breakdown at
+around 200 a.u. of time, the norm of the AB partition
+of the cluster amplitudes grows very rapidly. Likewise,
+the left vector norm of the regular TDCC calculation is
+also small at the start of the propagation, and blows up
+at around 200 a.u. of time. We have performed calcula-
+tions with various separations or the two helium atoms
+up to 1 × 106 ˚A, and the solutions still display the same
+behavior.
+Analytically, the TDCCSD observables should have
+the correct scaling behavior, as demonstrated in Sec-
+tion II C, but this is clearly not the case in our simu-
+lations. We argue that this discrepancy is due to two
+effects:
+the growth of the AB partition of the time-
+dependent amplitudes due to the sensitivity to small de-
+viations from zero in the AB partition of the initial clus-
+ter amplitudes and time-dependent Hamiltonians in nu-
+merical simulations, and the blowup of the AB partition
+of the left vector.
+In the third and fourth column of Fig. 2, results from
+modified TDCC simulations of two helium atoms are
+shown, where one atom is placed at the origin and the
+other at 1000 ˚A on the x-axis.
+For the results in the
+third column, the initial values and derivatives of the
+cluster amplitudes in the AB partition are set to zero.
+The excitation energy and norms of the remaining am-
+plitudes still blow up around 200 a.u. of time. Note that
+the cluster amplitudes are independent of the left vector
+amplitudes, and the cluster amplitudes in the AB par-
+tition would blow up regardless of the value of the left
+vector elements. For the results in the fourth column,
+the initial values and derivatives of the left vector ampli-
+tudes in the AB partition, which do not enter in TDCC
+expectation value expressions, are also set to zero. In
+this case, the numerical integration completes, and the
+two-helium results are equal to the single-helium results
+apart from the scaling factors of 2 and
+√
+2 for the energy
+and single-subsystem norms, respectively.
+
+7
+t (a.u.)
+1 He
+2 He
+2 He, tAB = 0
+2 He, tAB = lAB = 0
+0
+200
+400
+0
+1
+2
+∆⟨H(t)⟩ (Ha)
+0
+200
+400
+0
+1
+2
+∆⟨H(t)⟩/2 (Ha)
+0
+200
+400
+0
+1
+2
+0
+200
+400
+0
+1
+2
+0
+200
+400
+0
+250
+500
+∥tA∥
+0
+200
+400
+0
+250
+500
+�
+∥tA∥2+∥tB∥2/
+√
+2
+0
+200
+400
+0
+250
+500
+0
+200
+400
+0
+250
+500
+0
+200
+400
+0.25
+0.50
+0.75
+∥lA∥
+0
+200
+400
+0.25
+0.50
+0.75
+�
+∥lA∥2+∥lB∥2/
+√
+2
+0
+200
+400
+0.25
+0.50
+0.75
+0
+200
+400
+0.25
+0.50
+0.75
+0
+200
+400
+10−12
+10−6
+100
+∥tAB∥
+0
+200
+400
+10−8
+10−1
+106
+∥lAB∥
+0
+200
+400
+10−8
+10−1
+106
+FIG. 2.
+TDCC simulations of one helium atom (1 He) and two non-interacting helium atoms (2 He) in a slowly ramped
+laser field. The two helium atoms are simulated with regular TDCC, but also with TDCC where the initial values and time
+derivatives of the cluster amplitudes in the AB partition are set to zero (tAB = 0), and TDCC where the initial values and
+time derivatives of both the cluster and left vector amplitudes in the AB partition are set to zero (tAB = 0, lAB = 0). In the
+top row of panels, the time-dependent energy differences ∆ ⟨H(t)⟩ = ⟨H(t)⟩ − E0 are shown, where the two-helium results are
+scaled by 1/2 and E0 is the ground state energy. In the next two rows, the norms of the amplitude and left vector partitions
+corresponding to an excitation of a single subsystem are shown, where the two-helium results are scaled by 1/
+√
+2. In the bottom
+panel row, the norms of the amplitude and left vectors corresponding to an excitation of two subsystems are shown.
+C.
+Simulating collective Rabi oscillations with
+TD-EOM-CC
+To numerically investigate the scaling properties of
+TD-EOM-CC, the interaction with the field is first cal-
+culated for two helium atoms placed at the origin and
+at 1000 ˚A on the x-axis.
+The excitation energy and
+norms are shown together with the single-helium re-
+sults in Fig. 3, where the same normalization factors
+has been used as for the TDCC results. The frequency
+of the oscillations in the scaled excitation energy and
+single-subsystem norms increases, and their magnitude
+decreases, as the number of helium atoms increases from
+one to two.
+The simulations are however numerically
+stable without the need for modifying the equations de-
+scribing the time-dependence of the state, in contrast to
+the TDCC simulations.
+To further investigate the scaling properties, the in-
+teraction with the field is also calculated for three to
+five helium atoms, and for one helium atom together
+with one to two beryllium atoms, where all atoms are
+placed 1000 a.u. apart on the x-axis.
+The sinusoidal
+function A sin(Ωt + ϕ) + C is least-squares fitted to the
+time-dependent energy between t = 25 optical cycles
+and t = 500 a.u.. The estimated Rabi frequencies Ω of
+the oscillating energy are shown for all calculations in
+Fig. 4. The frequencies increase with the number of non-
+interacting subsystem. For the purposes of quantifying
+the scaling properties of the Rabi frequencies, the figure
+also includes the function A√ne + C least-squared fitted
+to the frequencies for one to five helium atoms, where
+ne is the number of electrons. The goodness of the fit
+
+8
+t (a.u.)
+1 He
+2 He
+0
+200
+400
+0
+1
+2
+∆⟨H(t)⟩ (Ha)
+0
+200
+400
+0
+1
+2
+∆⟨H(t)⟩/2 (Ha)
+0
+200
+400
+0.0
+0.5
+∥rA∥
+0
+200
+400
+0.0
+0.5
+�
+∥rA∥2+∥rB∥2/
+√
+2
+0
+200
+400
+0
+1
+∥lA∥
+0
+200
+400
+0
+1
+�
+∥lA∥2+∥lB∥2/
+√
+2
+0
+200
+400
+0.00
+0.05
+0.10
+∥rAB∥
+0
+200
+400
+0.00
+0.25
+0.50
+∥lAB∥
+FIG. 3.
+TD-EOM-CC simulations of one helium atom (1
+He) and two non-interacting helium atoms (2 He) in a slowly
+ramped external field. In the first row of panels, the time-
+dependent energy differences ∆ ⟨H(t)⟩ = ⟨H(t)⟩ − E0 are
+shown, where the two-helium results are scaled by 1/2 and
+E0 is the ground state energy.
+In the next two rows, the
+norms of the right and left vector partitions corresponding
+to an excitation of a single subsystem are shown, where the
+two-helium results are scaled by 1/
+√
+2. In the fourth panel
+row, the norms of the right and left vectors corresponding to
+an excitation of two subsystems are shown.
+demonstrates that the frequency increases as the square
+root of the total number of helium atom electrons. As the
+number of non-interacting subsystems in resonance with
+the field increases, we can expect the frequency to falsely
+appear to approach infinity, meaning that the method
+gives a qualitatively incorrect representation of interac-
+tions occurring at multiple sites simultaneously. The fig-
+ure also includes a constant C fitted to the Rabi fre-
+quencies calculated with the helium atom and one to two
+beryllium atoms. The goodness of the fit illustrates that
+the frequency does not scale with the number of systems
+that are not in resonance with the field. The representa-
+tion of the helium Rabi oscillation is therefore unaffected
+2
+4
+6
+8
+10
+ne
+2.0
+2.5
+3.0
+3.5
+4.0
+Ω (a.u.)
+×10−2
+1–5 He
+(1.30√ne + 0.08)×10−2
+1 He and 0–2 Be
+1.91×10−2
+FIG. 4. TD-EOM-CC calculations of helium and beryllium
+atoms in a slowly ramped laser field. The TD-EOM-CC Rabi
+frequencies Ω, obtained by least-squares fitting the function
+A sin(Ωt + ϕ)+C to the time-dependent energy, are given for
+different numbers of electrons in the system ne. The function
+A√ne+C is least-squares fitted to the time-dependent energy
+for 1–5 effectively non-interacting helium atoms, giving A =
+1.30 × 10−2 and C = 8 × 10−4. The constant C is fitted to
+the Rabi frequencies for 1 helium atom and 0–2 beryllium
+atoms, all non-interacting, giving C = 1.91 × 10−2.
+by the beryllium atoms, suggesting that TD-EOM-CC
+can represent interactions occurring at a single site of an
+extended quantum system.
+V.
+CONCLUSION
+In this work, we have demonstrated that the TDCC
+and TD-EOM-CC parametrizations can be expressed in a
+unified theoretical framework, where the time derivatives
+of the cluster amplitudes are taken as auxiliary condi-
+tions. We have furthermore implemented the TD-EOM-
+CC method in the elementary basis, and compared the
+scaling properties of the TDCC and TD-EOM-CC meth-
+ods through simulations of collective Rabi oscillations.
+We noted that the truncated TD-EOM-CC method fails
+to give a qualitatively correct representation of collective
+Rabi oscillations, as the Rabi frequency increases with
+the number of subsystems that are in resonance with the
+external field. However, we did not encounter any numer-
+ical instabilities in the TD-EOM-CC simulations, and the
+addition of subsystems that are not in resonance with the
+field did not affect the time-dependent energy. This indi-
+cates that TD-EOM-CC is suitable for simulating Rabi
+oscillations that occur at a single site. Although the in-
+troduction of additional Rabi oscillating subsystems neg-
+atively impacts the numerical stability of TDCC simula-
+tions, leading to solution blowup, the initial stages of the
+simulations display the correct scaling properties, as pre-
+dicted by the theory in Section II C. This supports the
+use of TDCC for simulating collective resonant excita-
+tions in extended systems, as long as the reference deter-
+
+9
+minant weight is not completely depleted. In conclusion,
+we propose that further research should be dedicated to
+the development of approximate methods that can give
+a qualitatively correct description of collective Rabi os-
+cillations without compromising numerical stability.
+ACKNOWLEDGMENTS
+This research has been financially supported by the Re-
+search Council of Norway through FRINATEK project
+nos.
+263110 and 275506, and computing resources
+have been provided by Sigma2 AS through project no.
+NN2962K. The authors would like to thank Alice Balbi
+for useful discussions.
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+Phys. 103, 2277 (2005).
+[20] T. B. Pedersen and S. Kvaal, J. Chem. Phys. 150, 144106
+(2019).
+[21] H. E. Kristiansen, Ø. S. Schøyen, S. Kvaal, and T. B.
+Pedersen, J. Chem. Phys. 152, 071102 (2020).
+[22] J. A. Sonk, M. Caricato, and H. B. Schlegel, J. Phys.
+Chem. A 115, 4678 (2011).
+[23] E. Luppi and M. Head-Gordon, Mol. Phys. 110, 909
+(2012).
+[24] A. S. Skeidsvoll, T. Moitra, A. Balbi, A. C. Paul, S. Co-
+riani, and H. Koch, Phys. Rev. A 105, 023103 (2022).
+[25] S. D. Folkestad, E. F. Kjønstad, R. H. Myhre, J. H. An-
+dersen, A. Balbi, S. Coriani, T. Giovannini, L. Goletto,
+T. S. Haugland, A. Hutcheson, I.-M. Høyvik, T. Moitra,
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+102, 023115 (2020).
+

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+arXiv:2301.01980v1  [nlin.CD]  5 Jan 2023
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+Fluctuation and Noise Letters
+© World Scientific Publishing Company
+Large Fluctuations in Amplifying Graphs
+Stefano Lepri
+Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi,
+Via Madonna del Piano 10 I-50019 Sesto Fiorentino, Italy
+[email protected]
+Received (received date)
+Revised (revised date)
+We consider a model for chaotic diffusion with amplification on graphs associated with
+piecewise-linear maps of the interval [S. Lepri, Chaos Sol. Fractals, 139,110003 (2020)].
+We determine the conditions for having fat-tailed invariant measures by considering ap-
+proximate solution of the Perron-Frobenius equation for generic graphs. An analogy with
+the statistical mechanics of a directed polymer is presented that allows for a physically
+appealing interpretation of the statistical regimes. The connection between non-Gaussian
+statistics and the generalized Lyapunov exponents L(q) is illustrated. Finally, some re-
+sults concerning large graphs are reported.
+Keywords: Chaotic map, Power-law distributions, Diffusion and amplification on graphs,
+Generalized Lyapunov exponents
+1. Introduction
+Large fluctuations are one of the distinctive features of complexity, being associated
+to lack of a characteristic scale and to extreme events.This work is part of a re-
+search program aimed at characterizing large fluctuations caused by the joint effect
+of energy diffusion and inhomogeneous amplification or growth. Diffusion can orig-
+inate from underlying disorder and scattering and/or chaotic motion, while growth
+stems for external energy pumping into the system. This leads to non-Gaussian
+fluctuations of the relevant physical quantities, whose statistical distributions can
+have fat-tails, leading to domination of a single event and lack of self-averaging of
+measurements [1]. This is well-known for multiplicative stochastic processes [2, 3]
+and chaotic dynamical systems that display intermittency and multifractality [4].
+A particularly interesting form of disorder is the one arising in dynamical sys-
+tems defined on graphs. They have many fascinating and diverse applications to
+describe complex interacting units with non-uniform connectivity [5]. When het-
+ereogeneous reaction is added a non trivial interplay between the connectivity and
+the local reaction emerges [6].
+Among the many possible physical examples, the example we mostly refer to is
+the one of active, disordered optical media where light amplification and scattering
+1
+
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+2
+Stefano Lepri
+coexist. [7]. This occurs in random lasers where indeed fat-tailed distributions of
+emission intensities are observed experimentally [8–13].
+This work reviews and extends some of the results of [14] were we introduced
+a simple dynamical system consisting of a map that couples chaotic diffusion and
+energy growth and dissipation. Nonlinear maps are time-discrete dynamical models,
+widely studied to establish the emergence of macroscopic behavior from microscopic
+chaos [15]. The model is inspired by experiments on lasing networks [16,17], con-
+sisting of active (lasing) and passive optical fibers supporting many optical modes,
+excited by external pumping. Optical coupling among the fibers provides a form
+of topological disorder and the system can be considered, loosely speaking, as a
+random laser on a graph. Another related experimental setup has been realized
+with nanophotonic devices by coupling a mesh of subwavelength waveguides [18].
+Heuristically, one may think of light as a bunch of rays undergoing chaotic diffusion
+and (site-dependent) amplification on such graph. As a matter of fact, the classi-
+cal dynamics of particles on graphs can be described by simple maps. Trajectories
+of a particle on a graph, undergoing scattering at its vertices, are in one-to-one
+correspondence with the ones of one-dimensional piecewise chaotic maps [19–21].
+The plan of the paper is as follows In Section 2 the recall and extend the
+map model introduced [14] along with some examples. In Section 3 we consider an
+approximate equation for the invariant measure and discuss the conditions for the
+appearance of the fat-tailed distributions. In Section 4 and examine the symmetries
+of the problem. Such conditions can be recasted in terms of a statistical mechanics
+problem: a polymer with a finite number of configurations in a random energy land-
+scape, as described in Section 5. A useful approach is based generalized Lyapunov
+exponents as discussed in Section 6. In Section 7 we report explicit analytical re-
+sults for the simplest case of a two-sites graph. Finally, we extend the analysis to
+examples of large graphs in Section 8.
+2. Graph with diffusion and amplification
+We consider the following map [14]
+�
+xn+1 = f(xn)
+En+1 = g(xn)En
+(1)
+where g(x) is positive and xn belongs to the unit interval. The function f is piecewise
+linear and we assume that the map is chaotic with a Lyapunov exponent λ1 > 0.
+The unit interval is partitioned in N disjoint intervals Ij of equal lengths 1/N and
+we consider a piece-wise constant gain function g,
+g(xn) = gj
+for xn ∈ Ij
+where the constants gj ≥ 1 and 0 < gj < 1 correspond to local amplification or
+dissipation respectively. Thus, the ”energy” variable En is coupled to xn, leading
+
+January 6, 2023
+1:30
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+ampligraph
+Large Fluctuations in Amplifying Graphs
+3
+to amplification fluctuations. Also, the sequence of multipliers g(xn) is in one-to-
+one correspondence with the symbolic dynamics of the map f and has the same
+time correlation in time. Maps of similar form have been considered in the context
+of on-off intermittency [22] and synchronization transition of two piecewise-linear
+chaotic maps [23].
+Assuming that the stationary invariant measure P(x, E) of the map is uniform
+in x, the Lyapunov exponents λ1,2 are computed straightforwardly
+λ1 =
+� 1
+0
+log |f ′(x)|dx,
+(2)
+λ2 = ⟨log(g(x))⟩ = 1
+N
+�
+j
+ln gj
+(3)
+Some specific examples are illustrated in Figure 1 along with their graph rep-
+resentation, constructed by examining the possible transitions in the underlying
+Markov dynamics. The first two examples f1, f2 depend on a parameter p (see
+Appendix) that controls the transition probabilities and have the same Lyapunov
+exponent λ1 = −p log p − (1 − p) log(1 − p). Note that λ1 > 0 but it is vanishingly
+small for p approaching 0 and 1 where the maps have weakly-unstable periodic
+orbits. The third example f3, has λ1 = log 3 and corresponds to the case of a com-
+plete four-sites graph where transition can occur towards any other site with the
+same probability. This example can be easily extended to arbitrary N (see Section
+8 below).
+In the stable case, λ2 < 0, the orbits tends to be attracted to the origin while
+λ2 > 0 they are repelled away and tent to grow indefinitely. In order to have a
+bounded invariant measure one needs to require that the variable En neither does
+drift to infinity nor is stucked at the origin. This can be implemented, for instance,
+by assuming that there are some ”barrier” points located at some prescribed values
+of E (the scale of E is arbitrary). This can be enforced deterministically: for instance
+for λ2 < 0 setting En+1 = s when En ≤ 0, where s is a small positive number. In
+this way, En is stationary and ranges in [0, ∞]. The quantity s is arbitrary, but we
+anticipate that the main results we are interested in do not depend on its value. a
+In the unstable case, λ2 > 0, we may impose the constraint at, say E = 1 resetting
+En+1 = 1 whenever En > 1. Another possibility would be to use stochastic or
+determinististic resetting, or to allow the trajectories to escape, see [14] for details.
+Starting from a ”Lagrangian” description in terms of chaotic trajectories we
+can derive the corresponding ”Eulerian” equations for the probabilities. The time-
+aAlso setting the variable to a new randomly chosen variable sn will do as well, as long as s is
+very small as it will only affect the shape of the invariant measure close to the boundaries [23].
+For a discussion of a similar problem for Langevin dynamics see Ref. [24] and the bibliography
+therein.
+
+January 6, 2023
+1:30
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+ampligraph
+4
+Stefano Lepri
+0
+0.5
+1
+xn
+0
+0.5
+1
+f1(xn)
+0
+0.5
+1
+xn
+0
+0.5
+1
+f3(xn)
+0
+0.5
+1
+xn
+0
+0.5
+1
+f2(xn)
+(a)
+1
+2
+1 − p
+p
+p
+1 − p
+(b)
+1
+2
+3
+4
+1 − p
+p
+p
+1 − p
+1 − p
+p
+1 − p
+p
+(c)
+1
+2
+3
+4
+Fig. 1. Left: three examples of the chaotic map and (right) their graph representations. The
+analytic expressions are give in the Appendix. Red and blue parts represents possible choices of
+amplifying (gj > 1) and dissipative (gj < 1) regions in phase space.
+discrete evolution of the measure Pn(x, E) is solution of Perron-Frobenius operator
+Pn+1(x, E) =
+�
+j
+1
+gj|f ′(yj)|Pn
+�
+yj, E
+gj
+�
+(4)
+where yj(x) = f −1(x) are the N pre-images of x. Boundary conditions are required
+to specify Pn(x, E) to take properly into account the barrier points.
+To give an idea of the dynamics, we report in Figure 2 some representative
+trajectories for the map f2 along with the attractors in phase spave and histograms
+of the variable z = log E. Note that an exponential decay at large z is a signature
+of a power-law tail in the variable E, that occurs when large fluctuations arise.
+3. Fast chaotic diffusion
+Since we are interested in the statistical properties of E, it is natural to consider, in
+view of definition of g(x), the probabilities Pj,n(E) to have a particle with energy
+E in each interval Ij. In general, it is not possible to write a closed equation for the
+Pj,n from (4). A simplified case is when the Lyapunov exponent λ1 is much larger
+then the typical rate of change of the energy variable. If so, we may assume that
+
+January 6, 2023
+1:30
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+ampligraph
+Large Fluctuations in Amplifying Graphs
+5
+Fig. 2. Time evolution and statistics of iterates of the map f2 given in Fig.1 for p = 0.95, 0.6, 0.01
+(bottom to top). The gain factors are g1,2,4 = 0.7, g3 = 2.7, corresponding to the Lyapunov
+exponent λ2 ≈ −0.0192. Left panels: trajectory snapshots, middle column: distribution of the
+iterates in phase space (xn, log En), histograms of the variable log En in semi-logarithmic scale.
+the measure becomes rapidly uniform in x within each interval Ij .We can thus look
+for solutions of (4) independent of x,
+Pi,n+1(E) =
+�
+j
+Wij
+gj
+Pj,n
+� E
+gj
+�
+(5)
+Then W is the N × N stochastic matrix for a random walk on a N-sites, directed
+graph. In our case the transition probabilities are given by the inverses of the
+map slopes (see the leftmost part of Figure 1 and the Appendix); W is symmetric
+and doubly stochastic, �
+j Wj,i = �
+j Wi,j = 1. This defines a Markov process in
+discrete time, as better seen by recasting (5) it the mathematically equivalent form
+Pi,n+1(E) =
+�
+j
+�
+Ki,j(E, E′) Pj,n(E′)dE′
+Ki,j(E, E′) = Wij δ(E − gjE′),
+that defines the transition rates Kij(E, E′) (this last equation reduces to (5) by
+integrating the δs). We also mention in passing that a Kramers-Moyal expansion,
+suitable when all the gj are close to unity, allows to derive a set of coupled Langevin
+
+200
+E
+100
+20
+n
+10
+10
+-10
+2
+4
+6
+80
+0.5
+110°10°104
+X
+n (10*)
+countsJanuary 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+6
+Stefano Lepri
+equations for the energies at each graph site. The details of the derivation will be
+reported elsewhere. Such equation contain stochastic advection terms, akin to the
+one found in the lattice case [25].
+4. Fat-tailed distributions
+Let us now examine the possibility of having sower-law solutions in the steady state
+Pi,n(E) = Pi(E) of the form Pi = QiE−β−1; with β > 0 for normalizability. The
+Qi are the marginal probabilities to be on site i with whatever energy. Substituting
+this Ansatz in the stationarity condition we obtain a consistency condition for β
+Qi =
+�
+j
+Wijgβ
+j Qj
+(6)
+i.e. Q must be an eigenvector of eigenvalue one of the matrix Wijgβ
+j , i.e.
+det(WGβ − 1) = 0,
+(7)
+where G is a diagonal matrix having elements gj. Note that β = 0 is always a
+solution. Moreover, the condition is invariant under the transformations
+gj −→ 1
+gj
+,
+β −→ −β.
+(8)
+This shoud be interpreted as follows. If there exist a distribution decaying for large
+E as E−β−1 in the stable case λ2 < 0, then the distribution in the unstable case
+with Lyapunov exponent −λ2 is Eβ−1 for small E (up to a cutoff set by the upper
+barrier).
+The more interesting regime occurs for |β| < 2 where the measures have di-
+verging variance (up to the barrier cutoff). Here, we expect a strongly intermittent
+dynamics, with dominance of single large fluctuations on the average, as in the
+well-know case of L´evy-stable distributions [26]. For a fixed W, the region in the
+N-dimensional parameter space (g1, . . . , gN) where this occurs, is thus bounded
+between the hyper-surfaces defined by the condition (7) with β = ±2 (see Section
+7 for an example).
+5. Statistical mechanics analogy
+Let us now show that finding a power-law decay of the measure can be interpreted
+as dual statistical mechanics problem. First, Equation (7) can be rewritten in an
+equivalent manner by imposing that the symmetrized matrix gβ/2
+i
+Wijgβ/2
+j
+has an
+eigenvalue equal to one. Let us define the quantities hj and Eij
+ln gj = λ2 + hj;
+Eij(β) = − 1
+β ln Wij − hi + hj
+2
+with �
+j hj = 0 and λ2 is defined by (3). We can thus rephrase the problem in terms
+of the statistical mechanics of a directed polymer of length ℓ whose microscopic
+
+January 6, 2023
+1:30
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+Large Fluctuations in Amplifying Graphs
+7
+configurations are labeled by sequences of σ1, σ2 . . . σℓ, with σi being an integer
+assuming values σi = 1, 2 . . ., N. The polymer energy is
+H =
+�
+i
+Eσi,σi+1.
+(9)
+The quantity Eij thus represents the energy cost between two consecutive beads of
+the polymer. It consist of two terms: the one dependent on W is a kind of elastic
+energy, while the hj represent some local energies, akin to the case of the polymer
+on a disordered substrate [27]. Larger positive values of hj correspond to stronger
+interaction with the substrate itself. b To obtain a thermodynamic state one has to
+impose some upper and lower bounds to the polymer energy in the same way done
+in the map model.
+The standard approach to compute the partition function associated with H, is
+to introduce the N × N transfer matrix
+T (σ1, σ2) = exp[−βEσ1,σ2];
+and β is interpreted as the inverse temperature . As it is well known, the partition
+function of the polymer of length ℓ is the trace of T ℓ or, equivalently, the sum of
+the eigenvalues τi of T , �
+i τ ℓ
+i . For large ℓ, we thus have to impose that its free
+energy, namely its the maximal eigenvalue is equal to exp(−βλ2),
+βλ2 = − log τ1.
+(10)
+This condition is indeed equivalent to (6) or (7). Notice that, considering the model
+parameters as fixed, this it is a kind of inverse procedure with respect to the stan-
+dard case: one fixes the free energy and wants to determine the corresponding tem-
+perature. By virtue of the Perron-Frobenius theorem since the matrix T is strictly
+positive then the leading eigenvalue τ1 is strictly positive and non degenerate. Also
+for a finite N it an analytic function of the element so there are no phase transitions.
+The analogy is also suggestive to understand the difference between the fat-
+tailed and Gaussian regimes. As said, the first case corresponds to 0 < β < 2.
+In the polymer language, this would corrrespond to the high-temperature regime
+where the elastic energy terms dominate on the pinning terms. In other words, the
+polymer is very stiff and the typical lowest-energy configurations will be trapped
+close to the largest hj. These configurations give large fluctuations above the average
+in agreement with the above point of view. On the contrary, for low temperatures
+the polymer is very loose, and explores the whole configuration space at low cost,
+making deviations from the average behavior very unlikely. In this respect the value
+β = 2 can be consider to define the characteristic temperature where the two energy
+terms balance.
+bOtherwise it could be represented as a chain of N-components spins. Here, the spin variables σi
+take values in the set 1, 2, . . . N on each site and hi is a spin-dependent constant magnetic field.
+In this interpretation it is reminiscent of the Potts model with nearest-neighbor interactions in
+1d. It is different from the simple standard case where the interaction is of the form δσi,σi+1.
+
+January 6, 2023
+1:30
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+ampligraph
+8
+Stefano Lepri
+Since τ1 > 0, equation (10) has no solution for λ2 > 0, and hence to thermo-
+dynamically stable states for the polymer problem in the canonical ensemble, as
+formulated so far. This correspond to the fact that for the dynamical system the
+origin is unstable. To account for this case, one can reason in two ways. One is to
+consider positive temperature states of a modified Hamiltonian −H, exploiting the
+symmetry (8). Alternatively, a more suggestive thermodynamic interpretation is in
+terms of negative absolute temperature. Let us consider the microcanonical states of
+the polymer with Hamiltonian H at total energy E. Than all the microscopic con-
+figurations are precisely those that reach such such energy. Upon increasing λ2 the
+number of such configurations, and thus the polymer entropy S(E), should decrease
+leading to negative temperature from the usual relation β = ∂S/∂E. Following the
+standard reasoning, we can thus regard the unstable regime as microcanonical state
+with negative absolute temperature where ensemble equivalence does not hold. [28].
+6. Generalized Lyapunov exponents
+A general and elegant approach to look at the problem of fat-tails is from the
+point of view of large deviation of Lyapunov exponents [31]. For the dynamical
+system like (1) one can consider the generalized Lyapunov exponents L(q), that
+are the growth rates of the qth moment of the perturbation as exp(L(q)τ) at large
+times τ [4,30,31]. The L(q) is the cumulant generating function of the associated
+variable and contains all the information on the fluctuations beyond the Gaussian
+regimes [32, 33]. The standard Lyapunov exponent is given by λ2 = L′(q = 0),
+and corresponds to the typical average growth of a fluctuation. Thus, deviations
+of L(q) from a linear behavior, λ2q, are a signature of intermittent dynamics [34].
+The existence of power-law stationary tails can be inferred from inspection of the
+behavior of the L(q) [23,35,36]. Indeed, if L(q) > 0 for large enough q then there is
+a finite probability for a small perturbation to grow very large with respect to the
+average. More precisely, the condition for power-law distributions with a diverging
+moments for q > q∗ is that L(q∗) = 0 [23,35]. Such condition must be equivalent to
+(7), namely a distribution decaying as E−1−q∗, i.e. q∗ = β. Also, the boundaries of
+the region with fat-tailed distribution are defined by L(±2) = 0.
+For the master equation (5), the generalized exponents can be computed exactly
+from equations (11). To this aim, we consider the equation without barrier boundary
+conditions and consider the moments of E in each portion Ii of the unit interval,
+ǫ(q)
+i,n ≡
+�
+EqPi,n(E)dE.
+By multiplying equation (5) by Eq and integrating in dE, we straightforwardly
+obtain a set of N difference equations
+ǫ(q)
+i,n+1 =
+�
+j
+Wijgq
+j ǫ(q)
+j,n
+(11)
+that are linear and closed at each order (moments of different order q are decoupled).
+Using the same notation as above, this amounts simply to compute the largest
+
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+Large Fluctuations in Amplifying Graphs
+9
+eigenvalue of the matrix WGq and evaluate L(q) as the logarithm of it, a procedure
+that is basically the same followed to compute the cumulant generating function
+of Markovian dynamics [37]. Note that the matrix can be made symmetric by the
+same transformation as given at the beginning of Section 5, so the eigenvalues are
+real. Also, comparing with (8), we see that the spectrum of generalized exponents
+is also invariant under the transformation gj → 1/gj and q → −q, which is related
+to the time-reversal invariance of the trajectories of the map.
+For general graphs the eigenvalues can be easily computed numerically. In Figure
+3 we report the exponents for the case of the map f2. It is known that the exponents
+are notoriously hard to compute expecially for large q values that requires sampling
+very unlikely trajectories [38,39]. We thus profit to test the accuracy of the direct
+method, with respect to the one based on computation of the eigenvalues . As seen
+from the data, for this simple example the direct method is in reasonable agreement,
+meaning that sampling accuracy is not a big issue in those examples.
+This approach is also accurate in reproducing the measured exponents ±λ2.
+For instance, as a numerical test for the case p = 0.6 of Figure 4, the condition
+L(q∗) = 0 yields q∗ ≈ 0.225.. to be compared from the fit of the distribution of
+z = log E yielding exp(−0.223z) for large z (see [14] for further numerical checks).
+-1
+0
+1
+2
+3
+q
+0
+0.2
+0.4
+0.6
+(b) p=0.8
+0
+1
+2
+q
+0
+0.2
+0.4
+0.6
+0.8
+L(q)
+(a) p=0.95
+-1
+0
+1
+2
+3
+q
+-0.04
+0
+0.04
+(c) p=0.05
+Fig. 3. The generalized Lyapunov exponents L(q) for the map f2 and for different values of
+parameter p and g = 1.2 l = 0.8. Symbols are the numerical values computed via the definition,
+for an ensemble of trajectories of finite duration t = 20 (a,b) and t = 160 (c); solid lines are the
+L(q) computed as the logarithm of the largest eigenvalue of the matrix W Gq (see text). In the
+case p = 0.95, we also draw the lines corresponding to the maximal and average growth rates,
+q log l and λ2q.
+In Figure 4a we compare two cases having parameters gj and 1/gj and thus
+opposite Lyapunov exponents. According to (8) the statistics of should have op-
+posite rates exp(±q∗z), as well verified by the data. It is also seen that the large
+fluctuation has a form of statistical symmetry, in the sense that their shape En for
+
+January 6, 2023
+1:30
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+ampligraph
+10
+Stefano Lepri
+λ2 < 0 would be similar to 1−En for λ2 < 0. (see Figure 4b) Moreover, rise and fall
+rates are accurately predicted by L′(q∗) and λ2, respectively, as read from Figure
+4c.
+-10 -8 -6 -4 -2
+log E
+10
+-4
+10
+-3
+10
+-2
+10
+-1
+10
+0
+PDF
+-1 -0.5 0
+0.5
+1 1.5
+q
+-0.005
+0
+0.005
+0.01
+0.015
+0.84
+0.86
+n (10
+6)
+10
+-4
+10
+-2
+10
+01.24
+1.26
+1.28
+10
+-4
+10
+-2
+10
+0
+(a)
+(b)
+(c)
+En
+L(q)
+Fig. 4. Comparison between the cases with opposite Lyapunov exponents ±λ2 for the map f2
+with p = 0.5, g1 = g3 = g4 = 0.9, g1 = 1.4 (orange curves) and g1 = g3 = g4 = 1/0.9, g1 = 1/1.4
+and (purple). Panel (a): distributions of zn = log En, the dashed line is the expected exponential
+behavior exp(−q∗z) where q∗ = −0.84 is the given by L(q∗) = 0. Panels (b): time series of En
+showing the build and decay of a large fluctuation: the dashed lines correspond to exponential
+growth/decay according to exp(λ2t) with λ2 = 5.09 10−3. Panel (c): the generalized Lyapunov
+exponents L(q) (solid line) for the case λ2 > 0, the dashed line is λ2q.
+7. Two-sites graph: analytical solutions
+For the simplest case of the two-sites graph, corresponding to the map f1 in Figure
+1 some explicit analytical results can be worked out. The stationary measure is
+solution of (letting g1 = g, g2 = l).
+P1(E) = p
+g P1
+�E
+g
+�
++ 1 − p
+l
+P2
+�E
+l
+�
+P2(E) = (1 − p)
+g
+P1
+�E
+g
+�
++ p
+l P2
+�E
+l
+�
+.
+(12)
+• The consistency conditions, Eq. (7) yields
+det(WGβ − 1) = −p(gβ + lβ) + (2p − 1)gβlβ + 1 = 0
+(13)
+where the transition matrix is given by W1 in (19) and Gβ ≡
+�gβ 0
+0 lβ
+�
+.
+The region in the (g, l) with large fluctuations |β| < 2 is bounded between
+the curves defined by (13) with β = ±2.
+
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+Large Fluctuations in Amplifying Graphs
+11
+• The statistical mechanics analogy can be worked out explicitely in the
+”spin” interpretation as an Ising chain. Let g = exp(λ2+h), l = exp(λ2−h)
+in (13), the transfer matrix concides with the well-known textbook expres-
+sion of the one-dimensional Ising model with β dependent parameters
+H =
+�
+i
+[−Jσiσi+1 + hσi]
+(λ2 ≡ λ). Upon letting
+p =
+eβJ
+eβJ + e−βJ ;
+1 − p =
+e−βJ
+eβJ + e−βJ ;
+J(β) = 1
+2β ln(
+p
+1 − p)
+condition (10) is rewritten in the familiar form
+exp(−βλ) = eβJ cosh βh +
+�
+e2βJ sinh2 βh + e−2βJ
+Also, it gives a nice interpretation of the parameter p. The very definition
+of J makes transparent that p is interpreted as a probability of a spin-flip
+and controls the type of interaction. In particular:
+– for p = 1/2: J = 0 and
+βλ = − ln cosh βh
+that correspond to 1d Ising paramagnet in external field h
+– For 1/2 < p < 1: J > 0 ferromagnetic interaction;
+– For 0 < p < 1/2: J < 0 antiferromagnetic interaction.
+In this language, the variable of interest is the magnetic energy of the spin
+chain.
+• Generalized Lyapunov exponents can be computed analytically as de-
+scribed above yielding [14]
+L(q) = log
+�����
+p(gq + lq) +
+�
+p2(gq + lq)2 − 4(2p − 1)gqlq
+2
+����� .
+(14)
+and it can be checked that the condition L(q∗) = 0 yields the same as (13).
+In Fig. 5 we summarize the various statistical regimes of the model, distinguish-
+ing the parameters values where fluctuations have diverging variance.
+8. Large graphs
+So far we have considered graphs with a small number of sites. A natural question
+would be how the result change upon increasing N, in particular whether the fat-
+tailed regimes persist. This is not an obvious question. For instance in large chaotic
+systems, the generalized Lyapunov exponents may become proportional to q. The
+heuristic explanation is that, due to fast correlation decay in spatio-temporal chaos,
+the norm vector is the sum of entries that grow almost independently [30]. In the
+
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+12
+Stefano Lepri
+0.8
+1
+1.2
+g
+0.8
+1
+1.2
+l
+β=−2
+β=2
+β=−1
+β=1
+Fig. 5. Phase diagram in the parameter plane (g, l) for two-sites graph, corresponding to the map
+f1 in Figure 1 with p = 0.6. The green line is the bifurcation line where λ2 = 0. There is an
+obvious reflection symmetry around the line g = l (black dashed). Shaded blue regions correspond
+to finite variance of the variable En yielding Gaussian fluctuations. The region bounded between
+the curves with β = ±2 is where L´evy-like fluctuations are expected.
+present example however the multipliers gj are quenched and the situation may be
+different.
+For simplicity, let us discuss directly the Markovian dynamics, Eq. (5). As a
+first instance, let us consider the ladder graph composed of N = 2M sites depicted
+in Fig. 6. For convention, we label the upper sites with even integers and the lower
+by odd ones. The transition matrix has non-zero elements given by
+W2i,2(i+1) = W2i+1,2i−1 = p;
+W2i,2i+1 = W2i+1,2i = 1 − p
+(15)
+for i = 1 . . . M, and Wij = 0 otherwise. Periodic boundaries have been assumed.
+This is a geometry which can be seen as an extension of the map f2 discussed above.
+2(i-1)
+2i
+2(i+i)
+2i-1
+2i+1
+2i+3
+1 − p
+1 − p
+1 − p
+1 − p
+1 − p
+1 − p
+p
+p
+p
+p
+p
+p
+p
+p
+Fig. 6. The ladder graph described by the transition matrix (15). Configuration with only one
+active site, labeled by red color.
+
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+Large Fluctuations in Amplifying Graphs
+13
+Another example is the complete graph
+Wi,i = 0;
+Wi,j =
+1
+N − 1
+i ̸= j
+(16)
+which is depicted in the lowest panel of Fig. 1 for N = 4. For simplicity, let us con-
+sider also the case of a single active site with all the other having equal dissipation,
+namely gj = g > 1 for a certain j = j0 and gj = l < 1 otherwise. With such choice,
+the Lyapunov exponent is λ2 = (1 − 1/N) log l + (log g)/N in both examples, and
+approaches the constant value λ2 ≈ log l < 0 for N large.
+In Fig. 7 we compare the generalized Lyapunov exponents (computed as above)
+for the two graphs for increasing size N. In the case of the ladder, the L(q) are
+N-independent and the example shown predicts that a fat-tail with q∗ ≈ 1 should
+persist upon increasing the size. On the contrary for the complete graph q∗ grows
+with N suggesting that the statistics should turn Gaussian for large enough N.
+-5
+0
+5
+10
+q
+-0,5
+0
+0,5
+1
+(b) complete graph
+-0,5
+0
+0,5
+1
+1,5
+2
+q
+-0,1
+-0,05
+0
+0,05
+0,1
+0,15
+L(q)
+N=16
+N=32
+N=64
+(a) ladder graph
+Fig. 7. The generalized Lyapunov exponents L(q) for (a) the ladder graphs and (b) the complete
+graph and different number of sites N; p = 0.4 g3 = 3.0 and gj = 0.7 for j ̸= 3.
+This is qualitatively consistent with the polymer interpretation, Equation (9).
+In the ladder case the elastic energy of the polymer is independent of the size. On
+the contrary for the complete graph the elastic energy decreases with N making
+the polymer more and more loose and hindering the observation of configurations
+associated with large energy fluctuations.
+9. Conclusion
+Motivated by experiments on active, disordered optical systems , we have studied
+a map model that combines chaotic diffusion and amplification on a graph [14].
+
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+14
+Stefano Lepri
+Within a stochastic approximation of the dynamics, given by the Markov process
+described by (5), we established the conditions for its invariant measure to display
+fat-tailed distributions in some regime of parameters. We mostly discussed some
+specific small graphs (N = 2, 4) examples and extended the results to large graphs
+(ladder and complete). The model has symmetries that allows to consider the stable
+and unstable cases in a simple way. Also, the problem can be interpreted in statisti-
+cal mechanics language, an analogy that can be fruitful to interpret the dynamical
+regimes. We have confirmed that the Generalized Lyapunov exponents provide a
+useful and simple tool to predict the fluctuation statistics and the anatomy of a large
+fluctuation, both in the stable and unstable cases. The model has its own interest
+but is also a guidance for interpretration of experiments in the optical disordered
+media as for instance the lasing networks [17].
+Acknowledgements
+The Author acknowledges S. Iubini, A. Politi and P. Politi for useful discussions
+and A. Di Garbo and R. Mannella for the invitation to the workshop DCP22 in
+Pisa.
+Appendix
+For reference we give here the functional forms of the examples considered in Fig.
+1:
+f1(x) =
+
+
+
+
+
+
+
+
+
+
+
+1
+px
+0 ≤ x ≤ p/2
+1
+1−px +
+1−2p
+2(1−p)
+p/2 < x ≤ 1/2
+1
+1−px −
+1
+2(1−p)
+1/2 < x ≤ 1 − p/2
+1
+px + 1 − 1
+p
+1 − p/2 < x ≤ 1
+(17)
+f2(x) =
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+x/p
+x < p/4
+(x − p/4)/(1 − p) + 1/4
+p/4 ≤ x < 1/4
+(x − 1/4)/p + 1/2
+1/4 < x < 1/4 + p/4
+(x − 1/4 − p/4)/(1 − p) + 3/4
+1/4 + p/4 < x < 1/2
+(x − 1/2)/(1 − p)
+1/2 < x < 3/4 − p/4
+(x − 3/4 + p/4)/p + 1/4
+3/4 − p/4 < x < 3/4
+(x − 3/4)/(1 − p) + 1/2
+3/4 < x < 1 − p/4
+(x − 1 + p/4)/p + 3/4
+1 > x > 1 − p/4
+(18)
+where 0 ≤ p ≤ 1. The stochastic matrices used in the text are
+W1 ≡
+�
+p
+1 − p
+1 − p
+p
+�
+,
+W2 ≡
+
+
+
+
+p
+1 − p
+0
+0
+0
+0
+p
+1 − p
+1 − p
+p
+0
+0
+0
+0
+1 − p
+p
+
+
+
+
+(19)
+
+January 6, 2023
+1:30
+WSPC/INSTRUCTION FILE
+ampligraph
+Large Fluctuations in Amplifying Graphs
+15
+while for f3 is given by (16) for N = 4.
+References
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+

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+Transition in vortex skyrmion structures in superfluid 3He-A driven by an analogue of
+the zero-charge effect
+R. Rantanen and V.B. Eltsov
+Department of Applied Physics, Aalto University, Finland
+(Dated: January 16, 2023)
+In quantum electrodynamics, the zero-charge effect originates from the logarithmic dependence
+of the coupling constant in the action of the electromagnetic field on the ratio of the ultraviolet and
+infrared energy cutoffs. An analogue of this effect in Weyl superfluid 3He-A is the logarithmic diver-
+gence of the bending energy of the orbital anisotropy axis at low temperatures, where temperature
+plays the role of the infrared cutoff and the vector of the orbital anisotropy plays the role of the
+vector potential of the synthetic electromagnetic field for Weyl fermions. We calculate numerically
+the spatial distribution of the order parameter in rotating 3He-A as a function of temperature. At
+temperatures close to the superfluid transition, we observe formation of vortex skyrmions known as
+the double-quantum vortex and the vortex sheet. These structures include alternating circular and
+hyperbolic merons as a bound pair or a chain, respectively. As temperature lowers towards absolute
+zero, we find a continuous transition in the vortex structures towards a state where the vorticity
+is distributed in thin tubes around the circular merons. For the vortex sheet, we present a phase
+diagram of the transition in the temperature – angular velocity plane and calculations of the nuclear
+magnetic resonance response.
+I.
+INTRODUCTION
+Superfluidity of helium-3 is realized in the spin-triplet
+p-wave pairing state [1]. The Cooper pairs have orbital
+momentum L = 1 and spin S = 1 and several distinct
+superfluid phases are found in the experiments [2]. The
+A phase, which is the focus of this work, is a chiral su-
+perfluid [3], where the components of a Cooper pair have
+equal spins, while all Cooper pairs have orbital momen-
+tum in the direction of the unit vector ˆl. The gap ∆ in
+the fermionic excitation spectrum of 3He-A is anisotropic
+and vanishes at two points on the Fermi surface along
+the orbital anisotropy axis defined by ˆl. Near these gap
+nodes the Bogoliubov excitations have properties of Weyl
+fermions [4].
+Weyl nodes lead to several types of anomalous be-
+haviour in 3He-A, including chiral anomaly [5–7], ther-
+mal Nieh-Yan anomaly [8], Bogoliubov Fermi surface and
+non-thermal normal component in moving 3He-A [9, 10],
+mass currents in the ground state [11, 12], and non-
+analytic coefficients in the expansion of free energy in
+terms of gradients of ˆl [13, 14].
+For Weyl fermions in
+3He-A, vector ˆl plays the role of the vector potential (up
+to a scaling factor), and thus its spatial variation and
+time dependence create a synthetic electromagnetic field.
+This effective electrodynamics possesses many features of
+the electrodynamics of quantum vacuum. In particular,
+the non-analyticity of the free energy is due to the loga-
+rithmic divergence of the coefficient Kb ∝ ln (∆/T), as-
+sociated with the term [ˆl × (∇ × ˆl)]2. This divergence is
+analogous to the running coupling constant in quantum
+electrodynamics [4], with the gap ∆ and the tempera-
+ture T playing the roles of the ultraviolet and infrared
+cut-offs, respectively [15]. The logarithmically divergent
+running coupling constant in QED is due to the screen-
+ing of electric charges by the polarized vacuum, known
+as the zero-charge effect.
+In liquid 3He-A, the spatial distribution of ˆl is rela-
+tively flexible and can be manipulated by an external
+magnetic field, solid boundaries and rotation.
+The A
+phase tends to respond to rotation by creation of a con-
+tinuous distribution of ˆl in the plane perpendicular to
+the rotation axis, formed from elements which carry both
+vortex and skyrmion topological charges, so-called vortex
+skyrmions. In this paper we present numerical calcula-
+tions on continuous vortex skyrmion structures at low
+temperatures, where the divergence of the bending coef-
+ficient Kb becomes relevant. The increased energy con-
+tribution from bending deformations of ˆl alters the vor-
+tex structures in a quantifiable manner. We have found
+a transition between two distinct core structures, and
+present a Ω-T phase diagram for the transition.
+In Sec. II we describe the different contributions to
+the free energy, and in Sec. III some possible realizations
+of vortex skyrmions in 3He-A: the double-quantum vor-
+tex and the vortex sheet. The numerical methods used
+to find the distribution of ˆl are described in Sec.
+IV.
+Section V briefly discusses the methods to calculate the
+nuclear magnetic resonance (NMR) response of the vor-
+tex skyrmion structures.
+The results of the paper are
+divided into four sections: in Sec.
+VI we present cal-
+culations of a model ATC vortex and quantitative pre-
+dictions on the low temperature structures, in Sec. VII
+results from vortex sheet calculations, in Sec. VIII from
+separate double-quantum vortices and finally in Sec. IX
+we compare the results with the predictions made in Sec.
+VI. The last section is dedicated to the conclusion.
+II.
+FREE ENERGY DENSITY OF 3He-A
+The order parameter in the A phase of superfluid 3He
+is separable in spin and orbital variables and has the
+arXiv:2301.05558v1  [cond-mat.other]  13 Jan 2023
+
+2
+form [1]
+Aµj = ∆A ˆdµ( ˆmj + iˆnj)
+(1)
+where unit vectors ˆm, ˆn and ˆl form an orthonormal triad
+with ˆl being the direction of the orbital angular mo-
+mentum of the Cooper pair, ˆd is a unit vector of spin
+anisotropy perpendicular to the preferred direction of
+the Cooper pair spin and ∆A is the temperature- and
+pressure-dependent maximum gap in the energy spec-
+trum of Bogoliubov quasiparticles.
+The orientation of the order parameter anisotropy axes
+is affected by multiple competing interactions.
+The
+dipole interaction between magnetic momenta of the
+atoms forming the Cooper pair results in spin-orbit cou-
+pling, with the free energy density
+fdip = 3
+5gd
+�
+1 − (ˆl · ˆd)2�
+.
+(2)
+The spin-orbit energy is minimized when ˆl is parallel or
+antiparallel to ˆd. The coefficient gd is expressed as [1]
+gd(T) = 4
+3λdN(0)∆A(T)2
+(3)
+where λd ≈ 5 × 10−7 is an approximately constant cou-
+pling parameter and N(0) is the pressure-dependent den-
+sity of states for one spin state.
+In an external magnetic field H, the spins of the
+Cooper pairs tend to align along it and thus ˆd favors
+orientation orthogonal to H.
+The magnetic (Zeeman)
+energy density is
+fmag = 1
+2∆χ
+�
+ˆd · H
+�2
+.
+(4)
+The coefficient ∆χ is the difference between the two
+eigenvalues of the spin susceptibility tensor [1]
+∆χ = 1
+2γ2ℏ2N(0) 1 − Y0
+1 + F a
+0 Y0
+(5)
+where γ = −20378 G−1s−1 is the gyromagnetic ratio of
+3He, Y0 is the Yosida function and F a
+0 ≈ −0.756 (at a
+pressure of 33 bar)[16] is the pressure dependent spin-
+asymmetric Landau parameter.
+Comparing Eqs. (4) and (2) one finds that the orien-
+tation effect of the magnetic field overcomes that of the
+spin-orbit interaction at the so-called dipolar field
+H∗ =
+�6
+5
+gd
+∆χ
+�1/2
+≈ 30 G.
+(6)
+In an isotropic superfluid such as 4He, the superfluid
+velocity vs is defined by the gradient of the phase φ of
+the order parameter ψ = |ψ|eiφ
+v(4He)
+s
+= ℏ
+m4
+∇φ.
+(7)
+In superfluid 3He-A, with an anisotropic order parameter,
+the situation is more complicated. The order parameter
+in Eq. (1) is invariant under relative gauge-orbit trans-
+formation and multiplying Aµj with a phase factor eiφ
+can be compensated by rotating ˆm and ˆn around ˆl by
+−φ, ie. by transforming ˆm + iˆn → e−iφ( ˆm′ + iˆn′). The
+phase of the order parameter is then intrinsically linked
+to its orientation through the orbital angular momentum
+vector ˆl. The superfluid velocity in the A phase is given
+by [17]
+vs =
+ℏ
+2m3
+�
+i
+ˆmi∇ˆni
+(8)
+where m3 is the mass of the 3He atom.
+Superflow is
+created by rotation of the orbital triad around a fixed
+ˆl, as well as by changes in the orientation of ˆl.
+The
+superfluid velocity in Eq. (8) is linked to the ˆl vector
+through the Mermin-Ho relation [18]:
+ω = ∇ × vs =
+ℏ
+4m3
+�
+ijk
+ϵijkˆli
+�
+∇ˆlj × ∇ˆlk
+�
+.
+(9)
+In the free energy we consider the terms with vs to be
+the kinetic energy. The kinetic energy density of 3He-A
+is
+fkin = 1
+2ρs⊥
+�
+ˆl × vs
+�2
++ 1
+2ρs∥
+�
+ˆl · vs
+�2
++ Cvs ·
+�
+∇ × ˆl
+�
+− C0
+�
+vs · ˆl
+�
+ˆl ·
+�
+∇ × ˆl
+�
+(10)
+where ρs⊥ and ρs∥ are the superfluid density in the di-
+rection perpendicular and parallel to ˆl, respectively, and
+C and C0 are coefficients related to the superflow. The
+first two terms in Eq. (10) can be written as
+1
+2ρs⊥v2
+s − 1
+2
+�
+ρs⊥ − ρs∥
+� �
+ˆl · vs
+�2
+.
+It is seen from here that it is energetically favorable for
+ˆl to be oriented along the superfluid velocity vs, since
+ρs⊥ > ρs∥ as the gap in the quasiparticle energy spectrum
+is at maximum in the direction perpendicular to ˆl while
+it is zero along ˆl.
+In a superfluid rotating with constant angular veloc-
+ity Ω, the normal fluid rotates as a solid body with the
+velocity
+vn = Ω × r.
+(11)
+For a rotating system, two extra terms are included in
+the free energy of the whole fluid,
+1
+2ρnv2
+n and −Ω · L
+where L is the total angular momentum. Adding these
+terms is equivalent to transforming vs → vs − vn in Eq.
+(10), when a constant term − 1
+2ρv2
+n is omitted.
+In the absence of an external magnetic field and ro-
+tation, the minimum energy configuration in the bulk
+corresponds to the uniform order parameter. This is due
+to the elastic energy associated with changes in the ori-
+entation of the ˆl and ˆd vectors
+
+3
+fel = 1
+2Ks
+�
+∇ · ˆl
+�2
++ 1
+2Kt
+�
+ˆl ·
+�
+∇ × ˆl
+��2
++ 1
+2Kb
+�
+ˆl ×
+�
+∇ × ˆl
+��2
++ 1
+2K5
+�
+a
+��
+ˆl · ∇
+�
+ˆda
+�2
++ 1
+2K6
+�
+a
+�
+ˆl × ∇ ˆda
+�2
+(12)
+where the terms with coefficients Ks, Kt and Kb corre-
+spond to splay-, twist-, and bend-like deformations in the
+ˆl-vector texture, respectively. The terms with K5 and K6
+are related to changes in the ˆd vector orientation.
+The temperature dependent coefficients Ki in the elas-
+tic energy are calculated using Cross’s weak coupling gas
+model [14], following Fetter [19] and using the Cross func-
+tions calculated by Thuneberg [20]. The coefficients en-
+tering in the free energy density are presented in Ap-
+pendix B. The bending coefficient Kb warrants special
+attention, as it is connected to the zero-charge effect in
+3He-A [4].
+It is logarithmically divergent as Kb(T) =
+Kb1 + Kb2 ln(Tc/T) when T → 0 owing to nodes in the
+energy gap in the spectrum of Bogoliubov quasiparticles
+[14].
+The boundary conditions imposed by the container
+walls are also crucial in determining the texture. Most
+importantly, the ˆl vectors are forced perpendicular to the
+boundary surface [21]. This means that the ˆl texture can-
+not be uniform in a system with finite size. In addition,
+the superflow through the walls must be zero [22], mean-
+ing that in the rotating frame vs − vn must be aligned
+parallel to the surface at the boundaries. Ignoring mag-
+netic relaxation effects near the surface, the spin currents
+and thus the gradients of the spin anisotropy vector com-
+ponents ∇ ˆda are aligned parallel to the boundaries.
+III.
+CONTINUOUS VORTICES
+The form of superfluid velocity in 3He-A, Eq. (8), al-
+lows for the formation of vortex structures that do not
+require the suppression of the amplitude of the order pa-
+rameter like in conventional superfluids and superconduc-
+tors. As shown by the Mermin-Ho relation (9), the vortic-
+ity ω can be non-zero in regions where ˆl is non-uniform.
+This means that non-singular vortices with continuous
+vorticity can exist in the superfluid.
+In this paper, we focus on continuous vortex structures.
+In 3He-A hard-core defects where the order parameter is
+suppressed are also possible. These types of structures
+are generally not formed when rotation is started in the
+superfluid state, due to their high critical velocity of nu-
+cleation compared to continuous vortices [23, 24].
+On a closed path around a vortex, the circulation is
+given by [25]
+νκ0 =
+�
+vs · dr =
+ℏ
+2m3
+S(ˆl)
+(13)
+where κ0 = h/2m3 is the quantum of circulation for 3He,
+ν is the number of circulation quanta and S(ˆl) is de-
+fined as the area on the unit sphere covered by the ori-
+entations of ˆl inside the domain bounded by the closed
+path.
+In Eq. (13), the first integral along the path is
+the usual expression of the topological invariant defining
+quantized vortices.
+The second integral over the area
+enclosed by the path is the topological invariant usu-
+ally used for defining skyrmions. The equivalence of the
+two expressions follows from the Mermin-Ho relation (9).
+The continuous vortex structures in 3He-A, surrounded
+by the volume where ˆl lies in a plane, possess both invari-
+ants, that is, they are simultaneously quantized vortices
+and skyrmions. The in-plane orientation of ˆl in the exter-
+nal regions, needed to ensure integer values of integrals
+in Eq. (13), is usually provided by the boundary condi-
+tions at the sample walls or by the combination of the
+spin-orbit (2) and Zeeman (4) interactions in the applied
+magnetic field.
+The simplest continuous vortex structure contains one
+quantum of circulation, and is known as the Mermin-
+Ho vortex. In the core of a Mermin-Ho vortex, the ˆl-
+vectors rotate out of the plane, covering exactly half of
+a unit sphere. Single Mermin-Ho vortices are observed
+in narrow cylinders [26] where they are stabilized by the
+effect of the container walls on the orientation of ˆl.
+A skyrmion in 3He-A is a topological object where the
+ˆl vectors cover the whole unit sphere, with ν = 2 quanta
+of circulation. An axisymmetric skyrmion known as the
+Anderson-Toulouse-Chechetkin (ATC) vortex [27, 28] is
+the simplest model for a double-quantum vortex in 3He-
+A. The ATC vortex with the axis along ˆz consists of a
+topological soliton separating a core region with ˆl = ˆz
+from an outer region with ˆl = −ˆz. In a finite system,
+however, the axisymmetry of the structure is broken by
+the bulk ˆl texture, which is confined to the xy plane
+by the boundary conditions at the walls parallel to ˆz.
+The non-axisymmetric double-quantum vortex, shown in
+figure 1a, consists of two merons: one circular and one
+hyperbolic. The circular meron covers the top half of the
+unit sphere and the hyperbolic meron the lower half. The
+vorticity ω in a double-quantum vortex (DQV) is concen-
+trated in a cylindrical tube around the axis of the vortex,
+with a vorticity free region between the two merons [29].
+These structures typically appear in systems with mag-
+netic fields above the dipolar field [24], where the order
+parameter is ”dipole-unlocked” inside the core, so that
+ˆd is forced to stay in-plane by the magnetic field while
+ˆl covers all possible directions. In lower magnetic fields,
+the merons arrange into a square lattice where a cell con-
+sists of two hyperbolic and two circular vortices, totaling
+four circulation quanta.
+At high rotation velocities, the vortex sheet is the pre-
+ferred texture over separate vortex lines [24]. A vortex
+sheet is a chain of alternating circular and hyperbolic
+
+4
+100 µm
+(a)
+0
+1e-3
+2e-3
+(b)
+40 µm
+40 µm
+(c)
+0
+0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
+0.01
+FIG. 1. Three different continous vortex structures with four
+quanta of circulation in 3He-A. The blue arrows represent
+the ˆl vector texture and the background color is vorticity
+in units of κ/µm2.
+(a) Two double-quantum vortices at
+Ω = 0.30 rad/s. (b) A circular vortex sheet at Ω = 5.70 rad/s.
+(c) Two separate vortex sheets, each with two quanta of cir-
+culation at Ω = 5.70 rad/s. The sheets are connected to the
+container walls outside the shown area.
+merons confined inside a topological soliton that sepa-
+rates two regions minimizing spin-orbit interaction en-
+ergy with one having ˆl ↑↑ ˆd and the other ˆl ↑↓ ˆd. A
+circular vortex sheet texture is shown in figure 1b. The
+sheet can be connected to the container walls (figure 1c),
+and as the rotation speed is increased, vortices enter the
+system through these connection points and the vortex
+sheet begins to spiral, meandering around the volume
+while keeping the soliton walls equidistant [30]. After a
+wall-connected soliton has appeared in the system, it be-
+comes difficult to nucleate separate vortices, as the criti-
+cal velocity of formation of new merons at the connection
+of the vortex sheet to the wall is lower than that of sep-
+arate DQVs [31, 32].
+IV.
+NUMERICAL METHODS
+We find the lowest-energy state of 3He-A in the London
+limit through numerical minimization. The calculation is
+done in two dimensions and we assume that the system is
+uniform in the z direction. The function to be minimized
+is the total energy per unit height F, defined as
+F =
+�
+S
+(fdip + fmag + fkin + fel) dS
+(14)
+The minimization is performed with respect to the spin
+anisotropy vector ˆd and the orbital triad consisting of
+the three orthonormal vectors ˆl, ˆm and ˆn. The param-
+eterization of the triad is done using unit quaternions.
+Quaternions have the benefit of reducing the number of
+variables from nine to only four, while also avoiding the
+problems associated with Euler angles like singularities
+and gimbal lock. The ˆd vector is parameterized with az-
+imuthal and polar angles α and β, where these problems
+are avoided by choosing the polar axis along the magnetic
+field direction. The magnetic energy (4) ensures that the
+polar angle should always be nonzero during the min-
+imization process in the dipole-unlocked regime we are
+interested in. The parameterization is presented in more
+detail in the Appendix.
+The calculations are done on two-dimensional circular
+domains with varying radii, which are meshed into tri-
+angular elements. The resolution used is limited by the
+available computational time and memory and varies be-
+tween 3.5 µm and 10 µm depending on the size of the sys-
+tem in question. The integration in Eq. (14) is done using
+Gaussian quadrature rules on the triangles. MATLAB is
+used to find the texture that minimizes the total energy
+using the BFGS method. The boundary conditions are
+implemented with the barrier method by adding an addi-
+tional energy term that penalizes parameter values that
+would violate the boundary conditions.
+The coefficients for the free energy terms are calculated
+at a pressure of 33 bar and varying temperatures. The
+magnetic field in the simulations is set to 0.55 T, as that
+is a high enough value for 3He-A to be stable down to
+zero temperature.
+V.
+NUCLEAR MAGNETIC RESONANCE
+Nuclear magnetic resonance (NMR) is a useful exper-
+imental tool for studies of superfluid helium-3. Different
+order parameter structures can usually be distinguished
+from the NMR absorption spectrum. In 3He-A, the long-
+range order of ˆl and ˆd together with the spin-orbit inter-
+action leads to spontaneously broken spin-orbit symme-
+try [33]. The coupling between the spin and orbital de-
+grees of freedom leads to an extra torque applied to the
+precessing spin in NMR experiments which allows us to
+probe the ˆl texture. Different vortex structures result in
+satellites in the NMR spectrum with characteristic fre-
+quency shifts [34]. We consider longitudinal NMR here,
+because at low temperatures we are interested in, 3He-A
+in bulk is stable at relatively high magnetic fields and the
+longitudinal resonance frequencies are independent of the
+magnetic field strength.
+Assuming a static equilibrium texture for ˆd = ˆd0, we
+parametrize the deviation of ˆd from the equilibrium due
+to the oscillating field with two parameters dH and dθ
+for the deviation along the field and perpendicular to the
+field, respectively. The ˆd vector in the presence of the
+
+45
+oscillating field is
+ˆd = ˆd0 + ( ˆ
+H × ˆd0)dθ + ˆ
+HdH
+(15)
+where ˆ
+H is a unit vector in the direction of the static
+magnetic field.
+The longitudinal NMR resonance fre-
+quencies are given by the Schr¨odinger-type equation [35]
+(D + U∥)dθ = α∥dθ
+(16)
+where the operator D is defined as
+Df = −5
+6
+K6
+gd
+∇2f − 5
+6
+K5 − K6
+gd
+∇ ·
+�
+ˆl(ˆl · ∇)f
+�
+(17)
+and the potential is
+U∥ = 1 − ( ˆ
+H · ˆl)2 − 2[ ˆ
+H · (ˆl × ˆd)]2
+(18)
+The resonance frequencies are related to the eigenvalues
+α∥ in (16) by
+ω2
+∥ = Ω2
+Aα∥
+(19)
+where ΩA is the Leggett frequency of the A phase.
+The NMR resonance frequencies are calculated by solv-
+ing the eigenvalue problem (16) using the finite element
+method (FEM). The same mesh from the energy min-
+imization is used and the equation is discretized using
+linear shape functions. The calculated eigenfunctions ψk
+are normalized so that
+�
+S
+|ψk|2dS = 1
+(20)
+for each eigenfunction. The convenience of FEM is that
+the method automatically enforces the Neumann bound-
+ary conditions for the spin waves. Dissipation effects are
+not taken into account in the NMR calculation, which
+may affect the results quantitatively.
+VI.
+ATC VORTEX
+At low temperatures, the logarithmic divergence of
+the Kb coefficient implies that the manifested struc-
+ture should be one that minimizes bending deformations.
+Since the ˆl vectors in a double-quantum vortex cover the
+whole unit sphere, this type of deformation cannot be
+avoided completely. In a circular ATC vortex, the ˆl vec-
+tors pointing along ˆz in the center are separated from an
+external region with ˆl along −ˆz by a topological twist
+soliton. There are two defining lengths for the soliton:
+the radius a and the thickness b, illustrated in the in-
+set in figure 2a. Along the radial direction there is only
+twist deformation, but on a loop around the vortex center
+ˆl bends (and splays) a full rotation, so that the elastic
+energy can be estimated as Fel ∼ Kb(b/a) + Kt(a/b).
+Therefore we expect that as the temperature decreases,
+the radius of the vortex increases, while the thickness of
+the soliton decreases. According to the Mermin-Ho re-
+lation (9), the vorticity ω is concentrated in the soliton,
+with no vorticity in the relatively uniform center.
+The structure suggested by Volovik [36] is this type of
+ATC vortex, where the ˆl texture in the soliton is
+ˆl = cos χ(r)ˆz + sin χ(r) ˆϕ
+(21)
+χ(r) = arccot
+�
+−r − a
+b
+�
+(22)
+where r, ϕ and z are the cylindrical coordinates. Follow-
+ing the derivation by Volovik, but including the effect of
+the rotation of the system in the kinetic energy gives us
+the following formulas for the soliton radius and thick-
+ness:
+a =
+�
+��
+ρ
+�
+ℏ
+m3
+�2
+2ρΩ ℏ
+m3 + 12
+5 πgd
+�
+Kt
+Kb
+�1/2
+�
+��
+1/2
+≈
+�
+ℏ
+2m3Ω
+�1/2
+(23)
+b = a
+�Kt
+Kb
+�1/2
+(24)
+The magnetic field H is transverse to the cylinder axis.
+To simplify the derivation, ˆd is assumed to be uniformly
+along ˆz and the terms with C and C0 in Eq. (10) have
+been ignored. At higher rotation speeds, however, these
+terms turn out to have a considerable effect.
+At finite rotation speeds, the temperature dependence
+of a is negligible, which gives the Kb-independent expres-
+sion in Eq. (23). Therefore the effects of the logarithmic
+divergence of Kb should only be seen in the narrowing of
+the domain wall.
+We numerically calculate the structure of the ATC vor-
+tex in a circular domain with a radius R = 115 µm. As
+the initial configuration for the energy minimization we
+set ˆl parallel to ˆz at the center and antiparallel to ˆz at
+the boundary, with a linear rotation around the radial
+direction in between. The boundary condition is applied
+such that ˆl stays antiparallel to the z-axis at the edge of
+the calculation domain. The magnetic field direction is
+along the y-axis and accordingly the ˆd vectors are uni-
+formly pointing in the z direction. The temperature is
+set to T = 0.005Tc. The angular velocity is initially set
+to the value Ω = 2.7 rad/s, as minimization with zero
+rotation results in the vortex drifting to the walls of the
+simulation box. The result of this minimization is then
+used as an initial state for the angular velocity sweep.
+An example of the ATC vortex texture is presented in
+figure 2a.
+The value of the angular velocity Ω is gradually de-
+creased in steps of 0.1 rad/s, starting from the initial
+value Ω = 2.7 rad/s. At each step of Ω, the previous
+minimization result is used as the new initial state, in
+order to mimic a realistic continuous deceleration. An
+
+6
+-100
+-50
+0
+50
+100
+x (µm)
+-100
+-50
+0
+50
+100
+y (µm)
+(a)
+0
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+5
+10-3
+a
+b
+(c)
+(d)
+(e)
+0
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+T/Tc
+0.2
+0.4
+0.6
+0.8
+1
+rel
+(b)
+c
+d
+e
+FIG. 2.
+(a) Spatial distribution (texture) of the orbital anisotropy axis ˆl (blue arrows) in the ATC vortex, where ˆl is parallel to
+the ˆz-axis in the center of the computational domain and antiparallel to it at the boundary. The color indicates the distribution
+of vorticity in units of κ/µm2. The predicted vorticity tube is clearly visible. (Inset) An example of a fit (line) of the angle χ
+determined from the simulation results along y = 0 (points) to Eq.(22). The radius a and thickness b are determined from the
+fit and illustrated in the figure. The calculations are performed at T = 0.01Tc and Ω = 2.3 rad/s. (b) The relative vorticity
+ωrel of the ATC vortex as a function of temperature. The values are calculated from the temperature sweep. (c)-(e) ATC
+vortex texture and vorticity distribution at temperatures 0.001Tc, 0.15Tc and 0.80Tc, respectively. The corresponding points
+are marked with filled black circles in (b). The color scale is the same as in (a).
+increasing Ω sweep is also performed, similarly starting
+at the initial Ω = 2.7 rad/s.
+The lowest energy during the Ω sweep is achieved at
+Ω = 2.3 rad/s.
+This value for Ω is used in the tem-
+perature sweep. The temperature sweep is started from
+T/Tc = 0.001 and the temperature is increased gradually
+in steps, ending at a temperature of T/Tc = 0.8. More
+points are calculated at lower temperatures T/Tc < 0.1,
+as that is the region where the logarithmic divergence of
+the Kb coefficient is relevant.
+From the simulation results we find the radius and
+width of the topological soliton by fitting values of
+cos−1(ˆlz) to the model χ(r) dependence in Eq. (22), with
+a and b as the fitting parameters. An example fit is shown
+in the inset in figure 2a. The fits are done along multi-
+ple radial lines going around the whole simulation disk,
+and the radius and width values a and b are taken as the
+mean of the fitting parameters over each line. There is
+slight axial asymmetry in the texture, due to the dipole
+interaction in the transverse field.
+The numerically calculated and predicted values of a
+and b for the ATC vortex are shown in figure 3. The pre-
+dicted dependence a−2 ∝ Ω and b−2 ∝ Ω is not so clear in
+the simulation results. A possible reason is the omission
+of the C and C0 terms in the kinetic energy in the model
+derivation. To test this possibility, a simulation with the
+same setup but without these terms has been performed,
+the results of which are also shown in figure 3.
+The analytic model is indeed in closer agreement with
+the simulation without C terms, especially at higher an-
+gular velocities. At low rotation speeds, the agreement
+becomes worse, as the numerically calculated structure
+becomes limited by the simulation domain.
+During the increasing temperature sweep, the vortic-
+ity distribution in the vortex becomes more uniform. Af-
+ter becoming completely uniform at around T = 0.15Tc,
+the tube distribution reappears, as shown in figures 2c-d.
+A good quantitative indicator for the presence of these
+vorticity tubes is found to be the relative vorticity ωrel
+defined as the ratio of the vorticity at the center of the
+vortex (where ˆlz = 1) to the maximum vorticity in the
+system
+ωrel =
+|ω|
+max(|ω|).
+(25)
+The relative vorticity for the ATC vortex is plotted in
+figure 2b. At both low and high temperatures ωrel ap-
+proaches zero, but the effect is caused by different interac-
+tions. At temperatures below T = 0.15Tc, the ˆl texture
+at the center of the vortex becomes more uniform and
+therefore vorticity-free due to the increasing energy cost
+of bending deformations. At high temperatures, the vor-
+tex center similarly becomes uniform, but this time as a
+result of the dipole interaction preferring the orientation
+ˆl ∥ ˆd = ˆz, due to the transverse magnetic field.
+As mentioned earlier, the axisymmetric ATC vortex
+
+7
+0
+0.02
+0.04
+0.06
+0.08
+T/Tc
+20
+30
+40
+50
+60
+Structure size ( m)
+0.5
+1
+1.5
+2
+2.5
+3
+3.5
+4
+4.5
+ (rad/s)
+0
+1
+2
+3
+4
+5
+Linearized structure size ( m-2)
+10-3
+anoC
+bnoC
+asimul
+bsimul
+apredict
+bpredict
+FIG. 3.
+Size of the vorticity tube in the ATC vortex. (Left) The width and radius of the vorticity tube in the ATC vortex,
+figure 2, as a function of temperature at Ω = 2.3 rad/s. The solid blue line and dashed red line correspond to the values of a
+and b calculated from Eqs. (23) and (24), respectively, and the filled blue circles (•) and red squares (■) are the corresponding
+values determined from the simulation results.
+The results from a simulation run where the kinetic energy terms with C
+and C0 coefficients have been set to zero are shown as empty blue circles (◦) and red squares (□) for a and b, respectively.
+The temperature dependence is similar in the numerical calculations and the model, although the radius is predicted to be
+significantly larger than numerically calculated. (Right) a−2 and b−2 as functions of rotation speed. From the equations (23)
+and (24), the behaviour is expected to be linear in these coordinates. The symbols are the same as in the left panel. The
+omission of the C terms provides a much better match for Ω-dependence. At low angular velocities, the discrepancy grows as
+the size of the computation domain limits the size of the vortex.
+is not the structure typically observed in realistic sys-
+tems. However, the simple model calculations indicate
+that the logarithmic divergence of Kb could cause textu-
+ral changes in more complicated vortex systems as well.
+The specific changes will depend on the structure in ques-
+tion, but the formation of topological twist solitons seems
+like a good candidate for reducing bending energy, if pos-
+sible.
+VII.
+VORTEX SHEET
+The first realistic structure we consider is the vortex
+sheet. The different merons inside a sheet are easily dis-
+tinguishable, and aside from the asymptotic behaviour
+the circular merons bear some similarities with the ATC
+vortex, which indicates that the bending energy could be
+reduced by a transformation like the one observed in the
+model vortex.
+The vortex sheet structure is constructed using vor-
+tex formation at a flow instability. The initial state at
+zero velocity is the so called PanAm-texture, an in-plane
+distribution of ˆl, where on one half of the sample circum-
+ference ˆl is directed inwards and on the other half out-
+wards. In simulations, reorientation of ˆl happens within
+a disretization triangle and the full texture includes two
+such defects. In real 3He-A, these defects have a hard
+core and thus cannot be adequately represented in our
+London-limit calculations. Nevertheless, on increase of
+the rotation velocity in simulations we observe that the
+defects act as a source of vorticity, as in the experiments
+[23], and this is sufficient for our purposes. The radius of
+the calculation domain is R = 115 µm, the magnetic field
+is applied along the y axis and the temperature is 0.80Tc.
+The angular velocity Ω is increased gradually in steps of
+0.1 rad/s, using the minimization result of the previous
+step as the initial state of the next one. After two DQVs
+enter the system, they merge into a circular sheet with
+four quanta of circulation like the one in figure 1b. Then
+Ω is decreased to find the rotation velocity where the
+total energy is lowest, which occurs at Ω = 5.7 rad/s.
+A detailed report on vortex formation and merging to
+sheets in our calculations will be published separately.
+With a stable four-quanta sheet, the temperature is
+decreased down to 0.006Tc.
+During the temperature
+sweep, the sheet reconnects with the container walls,
+splitting into two separate sheets (see figure 1c), each
+with two quanta of circulation embedded in a splay soli-
+ton. In order to test the stability of the newly formed
+two-sheet texture, further temperature sweeps were per-
+
+8
+x
+y
+(d)
+50 µm
+(a)
+(g)
+(b)
+(h)
+(c)
+(i)
+(e)
+0
+0.01
+(f)
+0
+1
+H
+FIG. 4. The zero-charge transition in the vortex sheet. The texture in (a-c) is calculated at 0.8Tc, the one in (d-f) at 0.2Tc
+and the one in (g-i) at 0.006Tc. (a) The ˆl-vector texture, shown using blue arrows, with the vorticity distribution given by the
+colormap in units of κ/µm2. The two vortex sheets are clearly visible, with vorticity distributed uniformly along the sheet. The
+centers of the circular and hyperbolic merons are marked with circles and crosses, respectively. (b) A close-up of the circular
+meron area marked by the dashed line in (a). The point where ˆlz = 1 is marked with a circle. (c) A density plot for the value
+of |ˆl × ˆd|. The regions where ˆl and ˆd are not aligned are darker. (d) The texture at 0.2Tc. The vorticity in the sheets has
+concentrated near the merons. (e) Close-up of the circular meron marked with a dashed line in (d). (f) The |ˆl × ˆd| density
+plot at 0.2Tc. The background soliton of the sheets is still there despite the change in vorticity distribution. (g) The texture
+at 0.006Tc, after the transition. The vorticity near the circular merons has formed the distinct tube shapes associated with
+the zero-charge effect. (h) A close-up of the circular meron marked with a dashed line in (g). (i) The |ˆl × ˆd| density plot at
+0.006Tc. The two sheets persist after the transition.
+formed starting from the lowest temperature of 0.006Tc
+back up to 0.80Tc and then down again. The two-sheet
+state persists.
+A transition similar to the one discussed in section VI
+occurs at temperatures below 0.05Tc. The textural tran-
+sition is shown in figure 4. At high temperatures (fig-
+ure 4a), the vortex sheet has the familiar structure with
+uniform vorticity along the sheet. As the temperature
+decreases, the vorticity becomes more concentrated at
+the merons (figure 4d).
+The vorticity forms ”bridges”
+between the circular and hyperbolic merons of the neigh-
+boring sheet, although the sheets still remain distinctly
+separate, as indicated by the |ˆl× ˆd| density plots in figure
+4c, f and i.
+Further decreasing the temperature to below 0.05Tc,
+the ˆl texture of the circular merons, marked with the
+dashed lines in figure 4, becomes more similar to the
+topological twist soliton. The meron center with vertical
+ˆl orientation increases in size, concentrating the bending
+deformation (and vorticity) into a tube. Comparing fig-
+ures 2d and 2c to figures 4e and 4h shows the similarities
+between the two transitions. The transition is smooth
+and shows no hysteresis on repeated temperature sweeps.
+A change can be observed in the vorticity near the
+hyperbolic merons as well. The vorticity at low temper-
+atures resembles a cross, with one line along the vortex
+sheet and one perpendicular to it. No quantitative anal-
+ysis has been performed on the hyperbolic meron struc-
+ture, but we believe that the change can be explained
+using the same reasoning as for the circular merons. The
+bending deformation in the hyperbolic meron is limited
+to the directions along these vorticity lines, while diago-
+nal to the ”arms” of the vorticity cross the ˆl vectors twist.
+In the hyperbolic merons the vorticity can be thought of
+as ”thin-sheet” vorticity instead of tube vorticity.
+The phase diagram in figure 5 is constructed by per-
+forming a temperature sweep for each rotation velocity.
+Initial states at different rotation velocities are found at
+T = 0.01Tc by changing Ω while simultaneously adjust-
+ing the radius of the cylinder accordingly to avoid vor-
+tices entering or escaping. Ω is then a measure of the
+vortex density of the system. The radius is changed by
+
+9
+0
+5
+10
+15
+20
+25
+30
+35
+ (rad/s)
+0
+0.02
+0.04
+0.06
+0.08
+T/Tc
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+T/Tc
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1
+rel
+ = 24.9 rad/s
+ = 3.67 rad/s
+ = 1.27 rad/s
+FIG. 5. (Left) An Ω-T phase diagram of the transition in the vortex sheet. The blue circles mark the transition temperatures
+at different values of Ω. The solid red line shows a fit of the expression A exp(−B/(Ω − Ω0)2/3) to the data points, with the
+A = 0.056, B = 0.657 (rad/s)2/3 and Ω0 = 1.15 rad/s. The red cross is the transition point calculated for a vortex sheet with 12
+quanta of circulation, and the filled circles indicate the points from the example fits on the right. (Right) Three example plots
+of relative vorticity ωrel as a function of temperature, taken from sweeps with Ω = 1.27, 3.67 and 24.9 rad/s. The calculated
+points are marked with triangles, diamonds and crosses, respectively, and the solid lines on each sweep show the two straight
+lines that best fit the data. The transition point is taken as the intersection of these fits and is marked with a circle in each
+sweep. The corresponding transition temperatures are approximately 0.0014Tc, 0.039Tc and 0.050Tc. The data from sweeps
+with lower Ω is more noisy, due to the reduced mesh resolution in the correspondingly larger cylinders.
+0
+0.2
+0.4
+0.6
+0.8
+T/Tc
+0.18
+0.2
+0.22
+0.24
+0.26
+0.28
+0.3
+0.32
+0.34
+0.36
+0.38
+||
+0.7
+0.8
+0.9
+1
+1.1
+1.2
+1.3
+1.4
+1.5
+o/
+x
+30 µm
+(b)
+(c)
+-1
+-0.8
+-0.6
+-0.4
+-0.2
+0
+0.2
+0.4
+0.6
+0.8
+1
+FIG. 6.
+(a) The eigenvalue α∥ of the largest satellite peak in the calculated NMR spectrum as a function of temperature
+plotted as blue circles and crosses. The circles correspond to the upward temperature sweep started from 0.006Tc while the
+crosses correspond to the return sweep back down from 0.80Tc. The near perfect match of the values of both sweeps indicates
+that there is no hysteresis in the transition. The solid black line is a logarithmic fit of the expression A + B ln(T/Tc + C) to
+the eigenvalues below 0.20Tc, with A = 0.277, B = 0.015 and C = 0.007. The red circles and crosses plotted against the right
+axis mark the ratio |ψo|/|ψx| of the magnitude of the eigenfunctions at the circular and hyperbolic meron centers. (b) The
+potential (18) for spin waves produced by the vortex sheet on the left side of the cylinder in figure 4d, at 0.20Tc. The circular
+and hyperbolic merons are difficult to distinguish visually. The centers of the circular and hyperbolic merons are marked with
+circles and crosses, respectively. (c) The potential for the vortex sheet in figure 4g, at 0.006Tc. There is clear asymmetry
+between the shapes of spin-wave traps at the two merons.
+
+10
+interpolating the texture to a cylinder with different size.
+Like for the model vortex, the relative vorticity ωrel is
+found to be a good quantitative indicator for the transi-
+tion.
+Above the transition temperature the maximum
+vorticity in the system is found at the centers of the
+merons and ωrel ≈ 1. Below the transition temperature,
+ωrel decreases linearly as the ˆl texture becomes more uni-
+form and the vorticity ω at the center of the circular
+meron decreases according to the Mermin-Ho relation
+(9).
+The linear decrease can be seen in the right side
+plot in figure 5.
+The transition temperature was found to be depen-
+dent on the vortex density of the system. A fit of the
+expression A exp(−B/(Ω − Ω0)2/3) to the data gives the
+asymptotic transition temperature at high rotation ve-
+locities T = 0.056Tc. (The origin of the exponents 2/3
+is discussed below.) At low velocities the transition tem-
+perature decreases and the zero temperature Ω cutoff is
+Ω0 = 1.15 rad/s.
+The calculation of the NMR response of the sheet at
+Ω = 5.7 rad/s as a function of temperature is shown in
+figure 6a. The eigenvalue α∥ of the most intense satel-
+lite peak in the NMR spectrum decreases linearly with
+temperature down to around T = 0.3Tc, after which the
+decrease becomes logarithmic α∥ ∝ ln(T/Tc). This in-
+dicates that the effects of the logarithmic divergence of
+the Kb coefficient could be observable through NMR ex-
+periments, although the required temperatures are very
+low.
+In the course of the transition the NMR potential dis-
+tribution (18) changes. At temperatures above the tran-
+sition the potential wells formed by the circular and hy-
+perbolic merons in the sheet look very similar, shown in
+figure 6b, while below the transition temperature there is
+a clear visual difference between the two (figure 6c) with
+a larger potential well at the hyperbolic meron. Corre-
+spondingly during the transition the eigenfunction be-
+comes more concentrated at the hyperbolic meron. The
+ratio between the magnitudes of the eigenfunction at the
+circular center |ψo| and at the hyperbolic center |ψx| is
+plotted in figure 6a. At higher temperatures the ratio
+|ψo|/|ψx| > 1 , while |ψo|/|ψx| < 1 at low temperatures.
+Notably the transition temperature seems to correspond
+roughly to the point where the ratio is close to one. This
+could be used as another indicator for the transition, al-
+though it is more indirect than the one used above.
+The transition to the tube vorticity state for the vortex
+sheet can be explained qualitatively using similar reason-
+ing that was used for the ATC vortex: the bending en-
+ergy is reduced by confining the deformations to a narrow
+tube around the center of the circular meron with uni-
+formly oriented ˆl. In the vortex sheet the largest relevant
+length scale for the ˆl vector gradients is the intermeron
+distance p along the sheet. The Ω−2/3 dependence of the
+transition temperature in the phase diagram of figure 5
+could be explained by the fact that p ∝ Ω−1/3 and above
+the transition the elastic energy density is proportional to
+p−2 ∝ Ω2/3. As the vortex density increases, the bending
+energy contribution from intermeron gradients becomes
+larger and can be reduced by the formation of vorticity
+tubes even at higher temperatures and smaller values of
+Kb. According to the fit in the phase diagram in figure 5,
+at T = 0 the transition occurs at a finite vortex density
+corresponding to Ω = 1.15 rad/s.
+VIII.
+DOUBLE-QUANTUM VORTICES
+The non-axisymmetric double-quantum vortex (see fig-
+ure 1a) is the most common topological object formed
+in 3He-A[37].
+The vorticity in a DQV is distributed
+in a tube around the vortex axis at all temperatures,
+so a priori it is difficult to determine what the qualita-
+tive effect the logarithmic divergence of the Kb coefficient
+would have on its structure. However, in the region be-
+tween the two merons the texture is similar to the ATC
+vortex. On the line between the hyperbolic and circu-
+lar meron centers, ˆl rotates with a twist-type deforma-
+tion over a distance d, while across the vortex (perpen-
+dicular to the line between merons) the deformation is
+splay/bend type over a distance w. The elastic energy is
+then Fel ∼ Kb(d/w) + Kt(w/d) and at low temperatures
+where Kb ≫ Kt, the energy is minimized by decreasing
+0
+0.05
+0.1
+0.15
+0.2
+0.25
+0.3
+0.35
+0.4
+T/Tc
+0
+0.2
+0.4
+0.6
+0.8
+rel
+T = 0.157Tc
+ = 0.30 rad/s
+(a)
+50 µm
+(c)
+50 µm
+(b)
+0
+FIG. 7.
+The transition in separate double-quantum vortices
+at Ω = 0.30 rad/s. (a) Plot of ωrel as a function of tempera-
+ture. The solid lines are the best linear fits to the data. The
+transition point T = 0.157Tc is taken as the intersection of
+these lines. (b) The ˆl-vector texture of the double-quantum
+vortex below the transition temperature at 0.001Tc. The color
+indicates vorticity ω. The centers of the circular and hyper-
+bolic meron are marked with a circle and a cross, respectively.
+(c) The texture at 0.20Tc.
+
+11
+0.01
+0.02
+0.03
+0.04
+0.05
+T/Tc
+0
+5
+10
+15
+20
+25
+30
+35
+Structure size (µm)
+10
+15
+20
+25
+30
+ (rad/s)
+0
+0.02
+0.04
+0.06
+0.08
+0.1
+0.12
+Linearized structure size (µm-2)
+asimul
+bsimul
+apredict
+bpredict
+FIG. 8.
+The radius a and width b of the circular meron vorticity tubes in the vortex sheet. The solid line and the dashed
+line are the predicted values of a and b, respectively, while the circles and squares correspond to their numerically calculated
+values. (Left) a and b as a function of temperature, taken from the sweep done at Ω = 11.55 rad/s. The value of Ω was chosen
+such that a wide enough temperature range was available below the transition temperature. The predicted values are much
+higher than calculated, almost six times higher for a and three times for b. The temperature dependence of b agrees with the
+prediction, while a decreases slightly with increasing temperature. (Right) a−2 and b−2 as functions of Ω at T = 0.01Tc. The
+calculated values are approximately linear in these coordinates, as predicted.
+the width d of the region between ˆl = ˆz and ˆl = −ˆz. In
+the whole DQV texture this is seen as a shift in the lo-
+cation of the vorticity tube, so that it is centered around
+the circular meron instead of the vortex axis. Then the
+size of the region around the meron center where ˆl is ori-
+ented along the vertical axis is again expected to increase,
+while the twist rotation occurs in a thin region between
+the two merons.
+A low-temperature state with two separated double-
+quantum vortices has been created as follows: a clean
+two-vortex state is found from an Ω sweep done in the
+axial field, starting with a PanAm-texture in a larger
+cylinder with radius R = 500 µm at 0.80Tc. The larger
+cylinder is chosen to keep applied rotation velocity be-
+low the treshold for merging vortices to sheets, and to
+accomodate the lower Ω values, the Ω sweep is done in
+smaller increments of 0.01 rad/s. When the first vortices
+have entered and found stable locations near the center
+of the cylinder at Ω = 1.27 rad/s, the temperature is re-
+duced down to 0.001Tc, the magnetic field rotated to the
+transverse direction and Ω reduced to 0.30 rad/s where
+the energy is at a minimum.
+A temperature sweep is performed in increasing direc-
+tion on this state with two double-quantum vortices. At
+the start of the sweep at 0.001Tc, the vorticity ω in the
+vortices is concentrated into a tube shape around the
+circular meron, with barely any vorticity near the hyper-
+bolic meron, as shown in figure 7b.
+The value of ωrel
+calculated at the center of the circular meron at these
+temperatures is close to zero.
+On increase of temperature, the value of ωrel increases
+linearly at low temperatures and stays constant above
+0.15Tc, as shown in figure 7a. Thus we find the transition
+of the same type as for the ATC vortex and the vortex
+sheet. However, the transition temperature for vortices is
+different from that for vortex sheets in the phase diagram
+in figure 5 at the same velocity. At such low velocities,
+the vortex sheet doesn’t appear to have a transition at
+all.
+Above the transition temperature the vortices look
+like well-known w-vortices with a hyperbolic and circu-
+lar meron (figure 7c). The vorticity is spread in a tube
+shape even at high temperatures, but the tubes are cen-
+tered around the axis of the whole two quanta structure.
+Below the transition point the tube shifts and becomes
+centered around only the circular meron (figure 7b).
+Qualitatively the low temperature texture resembles
+the ATC vortex: the core region of the circular meron is
+highly uniform, in order to minimize the region where ˆl
+bends. In a finite-radius cylinder, however, the ˆl vectors
+far from the vortex are horizontal instead of vertical, due
+to the orienting effect of the container walls.
+
+12
+IX.
+COMPARISON WITH THE ATC VORTEX
+The appearance of the tube vorticity distribution in
+the circular merons in vortex sheets and double-quantum
+vortices agrees qualitatively with the model ATC vortex
+in section VI. However, the quantitative prediction for
+the size of the tubes does not match well, as shown in
+figure 8 for the vortex sheet and in figure 9 for separated
+vortices. For sheets, the value of a is almost six times
+lower than predicted, while b is almost three times lower.
+In vortices the numerically calculated values differ by ap-
+proximately a factor of two from the predicted values.
+The lower measured values can be at least partially ex-
+plained qualitatively. The full simulations include the C-
+terms omitted in the model derivation, which were found
+to be highly impactful in section VI. Additionally, the
+ATC vortex structure in the model assumed an axisym-
+metric structure with the bulk ˆl texture being uniformly
+vertical. In realistic situations, the finite size of the do-
+main restricts the bulk texture to be in-plane due to the
+effects of the boundary conditions. In this case the bend-
+ing energy density cannot be strictly concentrated into a
+narrow tube, because outside the meron core there will be
+some splay/bend distortion in the bulk texture. Finally,
+the repulsive effect of neighboring quanta of circulation
+is expected to reduce the size of the tube by a factor that
+is dependent on the distance between quanta.
+In the vortex sheet, the tubes form around the circu-
+lar merons, which are single circulation quantum struc-
+tures instead of the ν = 2 ATC vortex considered in the
+model. The adjustment in the model equations (23) for
+a and (24) for b is done naively by assuming a superfluid
+velocity outside the vortex is twice smaller than in the
+original derivation. As mentioned previously, the change
+in number of quanta has an additional indirect effect on
+the calculated values through the change in the asymp-
+totic behaviour of ˆl vectors outside the vortex (horizontal
+vs. vertical).
+X.
+CONCLUSION
+We have numerically calculated equlibrium order-
+parameter textures in rotating 3He-A at low tempera-
+tures where the effect of the logarithmic divergence of the
+bending coefficient Kb in the free energy is relevant. The
+connection of this divergence to the zero-charge effect of
+quantum electrodynamics and the appearance of vortic-
+ity tubes at low temperatures was predicted by Volovik
+[36]. A transition to the predicted state has been found
+both in the vortex sheet and in separate vortices. The
+transition temperature is found to depend on the vortex
+density of the system, and a temperature-vortex density
+phase diagram has been presented for the vortex sheet.
+In our calculations in the absense of pinning, vortices
+are stable only with applied rotation. The original pre-
+diction of Ref. [36] has been adjusted to include the effect
+of rotation. The analytic model, nevertheless, does not
+0
+0.01
+0.02
+0.03
+0.04
+0.05
+T/Tc
+0
+50
+100
+150
+200
+Structure size (µm)
+asimul
+bsimul
+apredict
+bpredict
+FIG. 9.
+The radius a and width b of the vorticity tubes
+in separated double-quantum vortices as a function of tem-
+perature. The predicted a and b are marked by a solid and
+dashed line, respectively. The measured values are marked
+with circles and squares for a and b, respectively. The pre-
+dicted values are higher by approximately a factor of 2.
+capture all the details of the realistic textures and the
+size of the vorticity tubes in the simulated textures is
+considerably smaller than that in the model. In partic-
+ular, the so-called C term in the superfluid velocity, ig-
+nored in the model, turned out to play an important role
+in shaping vortex structures. Another important differ-
+ence between the model and the realistic textures is the
+asymptotic behaviour of ˆl at large radii, where the model
+ignores solid-wall boundary conditions. The calculations
+also have their limitations: They are done with the as-
+sumption of a uniform texture in the z direction, which
+means that possible three dimensional structures, related
+to the axial superflow in broken-symmetry vortex cores,
+could not be found.
+In search of observable signatures of the transition, we
+have calculated the NMR response of the vortex sheet
+as a function of temperature. As expected, restructur-
+ing of the distribution of vorticity has a profound effect
+on the frequency shift of the characteristic satellite in
+the NMR spectrum: The satellite moves further from
+the bulk peak towards the Larmor value. The logarith-
+mic dependence of the frequency shift, reflecting that of
+Kb becomes prominent only at temperatures below 0.2 Tc
+which may make observation of this effect in experiments
+challenging.
+ACKNOWLEDGMENTS
+We thank Grigory Volovik, Erkki Thuneberg and
+Jaakko Nissinen for stimulating discussions. This work
+
+13
+has been supported by the European Research Coun-
+cil (ERC) under the European Union’s Horizon 2020 re-
+search and innovation programme (Grant Agreement No.
+694248) and by Academy of Finland (grant 332964).
+We acknowledge the computational resources provided
+by the Aalto Science-IT project.
+Appendix A: Parameterization of the order
+parameter
+Quaternions are an extension of the complex number
+system into four dimensions and are of the form
+q = q0 + q1i + q2j + q3k
+(A1)
+with imaginary units i, j and k defined by the relation
+i2 = j2 = k2 = ijk = −1.
+(A2)
+Sometimes it is useful to use the notation
+q = q0 + q
+(A3)
+where q0 is called the real part and q the vector part.
+Three dimensional rotations and orientations can be
+described by quaternions, analoguously to how complex
+numbers can be used to represent two dimensional rota-
+tions. A rotation in 3D defined by an unit vector axis u
+and an angle θ can be expressed as a unit quaternion
+q = cos θ
+2 + sin θ
+2u
+(A4)
+The orientation of the orthonormal orbital triad
+( ˆm, ˆn, ˆl) can be represented with a single quaternion us-
+ing the conversion formula for rotation matrices:
+�
+�
+mx nx lx
+my ny ly
+mz nz lz
+�
+�
+=
+�
+�
+1 − 2q2
+2 − 2q2
+3
+2(q1q2 − q3q0) 2(q1q3 + q2q0)
+2(q1q2 + q3q0) 1 − 2q2
+1 − 2q2
+3
+2(q2q3 − q1q0)
+2(q1q3 − q2q0) 2(q1q0 + q2q3) 1 − 2q2
+1 − 2q2
+2
+�
+�
+(A5)
+The benefit of quaternions over other rotation for-
+malisms is that they reduce the number of required pa-
+rameters from nine to four, and they can describe any
+orientation without singularities or gimbal lock.
+The spin anisotropy vector ˆd is parameterized with
+azimuthal and polar angles α and β.
+To avoid issues
+when β = 0, the polar axis is chosen as the magnetic field
+direction H. In our system the H vector is confined to
+the yz plane and its direction is described by an angle
+µ between H and the z-axis. The ˆd vector can then be
+parameterized as
+dx = cos α sin β
+dy = cos β sin µ − cos µ sin α sin β
+(A6)
+dz = cos β cos µ + sin α sin β sin µ
+The magnetic field direction is kept static during mini-
+mization, so µ is a constant.
+The quaternion and ˆd angle values are defined at each
+node in the mesh.
+To calculate the energy for a sin-
+gle triangle, the parameters are linearly interpolated to
+quadrature points using barycentric coordinates, where
+the energy densities are computed.
+The integration is
+then performed using Gaussian quadrature rules. After
+the quaternions are interpolated, they must be renormal-
+ized to keep them unit length.
+Appendix B: Coefficients
+The coefficients for the energy densities in Equations
+(2), (4), (10) and (12) are presented in Figure 10. The
+values are normalized to ρ∥ in order to demonstrate their
+relative behaviour. Note the logarithmic divergence of
+the bending coefficient Kb as T → 0.
+0
+0.5
+1
+T/Tc
+0
+0.5
+1
+1.5
+2
+2.5
+3
+||
+s
+Kb
+C0
+K6
+K5
+C
+Kt
+Ks
+FIG. 10.
+The coefficients values used in the energy cal-
+culation as a function of temperature. The coefficients are
+normalized to ρ∥ to better visualize their relationships.
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+

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+arXiv:2301.11244v1  [math.OC]  26 Jan 2023
+ANOTHER LOOK AT PARTIALLY OBSERVED OPTIMAL
+STOCHASTIC CONTROL: EXISTENCE, ERGODICITY, AND
+APPROXIMATIONS WITHOUT BELIEF-REDUCTION
+SERDAR YÜKSEL ∗
+Abstract. We present an alternative view for the study of optimal control of partially observed
+Markov Decision Processes (POMDPs).
+We first revisit the traditional (and by now standard)
+separated-design method of reducing the problem to fully observed MDPs (belief-MDPs), and present
+conditions for the existence of optimal policies. Then, rather than working with this standard method,
+we define a Markov chain taking values in an infinite dimensional product space with control actions
+and the state process causally conditionally independent given the measurement/information process.
+We provide new sufficient conditions for the existence of optimal control policies. In particular, while
+in the belief-MDP reduction of POMDPs, weak Feller condition requirement imposes total variation
+continuity on either the system kernel or the measurement kernel, with the approach of this paper
+only weak continuity of both the transition kernel and the measurement kernel is needed (and total
+variation continuity is not) together with regularity conditions related to filter stability.
+For the
+average cost setup, we provide a general approach on how to initialize the randomness which we
+show to establish convergence to optimal cost.
+For the discounted cost setup, we establish near
+optimality of finite window policies via a direct argument involving near optimality of quantized
+approximations for MDPs under weak Feller continuity, where finite truncations of memory can be
+viewed as quantizations of infinite memory with a uniform diameter in each finite window restriction
+under the product metric. In the control-free case, our analysis leads to new and weak conditions for
+the existence and uniqueness of invariant probability measures for non-linear filter processes, where
+we show that unique ergodicity of the measurement process and a measurability condition related to
+filter stability leads to unique ergodicity.
+AMS subject classifications: 60J20,60J05,93E11
+1. Introduction, Literature Review, and Main Results. Partially ob-
+served Markov Decision processes (POMDPs) present challenging mathematical prob-
+lems with significant applied relevance. It is known that any POMDP can be reduced
+to a (completely observable) MDP [69], [51], whose states are the posterior state dis-
+tributions or beliefs of the observer; in particular, the belief-MDP is a (fully observed)
+Markov decision process.
+However, the use of belief-MDPs requires one to establish several regularity, con-
+tinuity and stability (such as filter stability or unique ergodicity) results for arriving
+at existence, finite memory approximations, robustness, as well as learning theoretic
+results (see e.g. [65, 39, 40, 48]. While significant progress on the regularity properties
+of belief-MDPs has been made in the literature, in this paper we will see that further
+refinements are possible if one does not restrict the analysis to a belief-MDP based
+formulation. In this paper, we present an alternative view for the study of infinite
+horizon average-cost or discounted cost optimal control of POMDPs. Our approach
+will be on a design which is not based on separation / or belief-MDP reduction. We
+define a Markov chain taking values in an infinite dimensional product space with
+control actions and the state process causally conditionally independent given the
+measurement process.
+For the controlled case, we provide new sufficient conditions for the existence
+of optimal control policies for average and discounted cost criteria, and under the
+latter criterion our analysis will establish a general result on near optimality of finite
+memory policies. In the control-free case, our analysis leads to sufficient conditions
+∗Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada, K7L
+3N6. Email: [email protected]. This research was partially supported by the Natural Sciences
+and Engineering Research Council of Canada (NSERC).
+1
+
+for the existence and uniqueness of an invariant probability measure for non-linear
+filters.
+We now present the problem. Consider a stochastic process {Xk, k ∈ Z+}, where
+each element Xk takes values in some standard Borel space X, with dynamics described
+by
+Xk+1 = F(Xk, Uk, Wk)
+(1.1)
+Yk = G(Xk, Vk)
+(1.2)
+where Yk is an Y-valued measurement sequence; we take Y also to be some standard
+Borel space. Suppose further that X0 ∼ µ. Here, Wk, Vk are mutually independent
+i.i.d. noise processes. This system is subjected to a control/decision process where
+the control/decision at time n, Un, incurs a cost c(Xn, Un). The decision maker only
+has access to the measurement process Yn and Un causally: An admissible policy γ is
+a sequence of control/decision functions {γt, t ∈ Z+} such that γt is measurable with
+respect to the σ-algebra generated by the information variables
+It = {Y[0,t], U[0,t−1]},
+t ∈ N,
+I0 = {Y0}.
+so that
+Ut = γt(It),
+t ∈ Z+
+(1.3)
+are the U-valued control/decision actions and we use the notation
+Y[0,t] = {Ys, 0 ≤ s ≤ t},
+U[0,t−1] = {Us, 0 ≤ s ≤ t − 1}.
+We define Γ to be the set of all such (strong-sense) admissible policies. We emphasize
+the implicit assumption here that the control policy can also depend on the prior
+probability measure µ.
+We assume that all of the random variables are defined on a common probabil-
+ity space (Ω, F, P). We note that (1.1)-(1.2) can also, equivalently (via stochastic
+realization results [26, Lemma 1.2] [9, Lemma 3.1]), be represented with transition
+kernels: the state transition kernel is denoted with T so that for Borel B ⊂ X
+T (B|x, u) := P(X1 ∈ B|X0 = x, U0 = u),
+.
+We will denote the measurement kernel with Q so that for Borel B ⊂ Y:
+Q(B|x) := P(Y0 ∈ B|X0 = x).
+For (1.1)-(1.2), we are interested in minimizing either the average-cost optimiza-
+tion criterion
+J(µ, γ) := lim sup
+N→∞
+1
+N Eγ
+µ[
+N−1
+�
+k=0
+c(Xk, Uk)]
+(1.4)
+or the discounted cost criterion (for some β ∈ (0, 1)
+J(µ, γ) := Eγ
+µ[
+∞
+�
+k=0
+βkc(Xk, Uk)]
+(1.5)
+2
+
+over all admissible control policies γ = {γ0, γ1, · · · , } ∈ Γ with X0 ∼ µ. With P(U)
+denoting the set of probability measures on U endowed with the weak convergence
+topology, we will also, when needed, allow for independent randomizations so that
+γn(In) is P(U)-valued for each realization of In. Here c : X × U → R+ is the cost
+function.
+We will also consider the control-free case where the system equation (1.1) does
+not have control dependence; in this case only a decision is to be made at every time
+stage; U is present only in the cost expression in (1.4). This important special case
+has been studied extensively in the theory of non-linear filtering.
+1.1. Literature review and preliminaries. It is well-known that any POMDP
+can be reduced to a (completely observable) MDP [69], [51], whose states are the pos-
+terior state probabilities, or beliefs, of the observer; that is, the state at time t is
+πt( · ) := P{Xt ∈ · |Y0, . . . , Yt, U0, . . . , Ut−1} ∈ P(X).
+We call this equivalent MDP the belief-MDP. The belief-MDP has state space P(X)
+and action space U. Here, P(X) is equipped with the Borel σ-algebra generated by
+the topology of weak convergence [6]. Since X is a Borel space, P(X) is metrizable
+with the Prokhorov metric which makes P(X) into a Borel space [49]. The transition
+probability η of the belief-MDP can be constructed as follows (see also [29]). If we
+define the measurable function
+F(π, a, y) := Pr{Xt+1 ∈ · |πt = π, Ut = u, Yt+1 = y}
+from P(X) × U × Y to P(X) and the stochastic kernel H( · |π, u) := Pr{Yt+1 ∈ · |πt =
+π, Ut = u} on Y given P(X) × U, then η can be written as
+η( · |π, u) =
+�
+Y
+1{F (π,u,y)∈ · }H(dy|π, u).
+(1.6)
+The one-stage cost function c of the belief-MDP is given by
+˜c(π, u) :=
+�
+X
+c(x, u)π(dx).
+(1.7)
+In particular, the belief-MDP is a (fully observed) Markov decision process with the
+components (P(X), U, η, ˜c).
+For finite horizon problems and a large class of infinite horizon discounted cost
+problems it is a standard result that an optimal control policy will use the belief πt
+as a sufficient statistic for optimal policies (see [69, 51, 8]).
+For the special case without control, the belief process is known as the (non-
+linear) filter process, and by the discussion above, this itself is a Markov process.
+The stability properties of such processes has been studied, where the existence of an
+invariant probability measure for the belief process, as well as the uniqueness of such
+a measure (i.e., the unique ergodicity property) has been investigated under various
+conditions, see. e.g. [15]. For the control-free case, [16] provides a comprehensive
+discussion on both the ergodicity of the filter process as well as filter stability, when
+the state space is finite but the measurement space is not necessarily so, under the
+further assumption that the unobserved state process is stationary and ergodic. [46,
+Theorem 2] and [61, Prop 2.1] assume that the hidden state process is ergodic and the
+filter is stable (almost surely or in expectation under total variation); these papers
+3
+
+crucially embed the stationary state in the joint process (xk, πk) and note that when xk
+is stationary, the Markov chain defined by this process admits an invariant probability
+measure. The unique ergodicity argument builds on the fact that any two invariant
+probability measures would have to have the same marginal invariant measure on the
+state process xk, and this leads to a direct argument in [61, Lemma B.1] to relate
+unique ergodicity to filter stability. In the context of finite state and measurement
+spaces, another line of work, adopted in [33],[24] and [16], studies the ergodicity and
+reachability properties of random matrix products, see [34] for a countable space setup.
+In [24] a reachability condition (across all initial priors) through the approximability
+of a rank-one matrix of unnormalized product of transition matrices, and in [33] a
+more restrictive subrectangularity condition, is utilized to establish unique ergodicity.
+A related argument appears in [59], in a general context. Finally, [16] established
+that the conditions of [24] are tight; with a different sufficiency proof related to filter
+stability and convex ordering of measures [58].
+For the controlled-setup, [50, 23, 52, 32] study the average-cost control problem
+under the assumption that the state space is finite; they provide reachability type
+conditions for the belief kernels. References [10] considers the finite model setup and
+[60] considers the case with finite-dimensional real-valued state spaces under some
+restrictive assumptions on the controlled state process leading to strong ergodicity
+conditions. In a related discussion, [21] studies the existence problem for the average-
+cost control problem under weak continuity conditions for the controlled kernels. In
+all of the aforementioned studies, the vanishing discount method is considered.
+Regularity of Belief-MDPs.
+As noted earlier, if one wishes to follow the
+traditional method of reducing a POMDP to a belief-MDP for studying the existence
+and structure of the optimal policies, it would be important to obtain continuity
+properties of the belief-MDP so that the standard measurable selection theorems, e.g.
+[30, Chapter 3] can be invoked to establish the existence of optimal control policies.
+Building on [35] and [22], we briefly review the weak Feller property of the kernel
+defined in (1.6) under two different sets of assumptions.
+Assumption 1.
+(i) The transition probability T (·|x, u) is weakly continuous in (x, u), i.e., for any
+(xn, un) → (x, u), T (·|xn, un) → T (·|x, u) weakly.
+(ii) The observation channel Q(·|x) is continuous in total variation, i.e., for any
+xn → x, Q(·|xn) → Q(·|x) in total variation.
+Assumption 2. The transition probability T (·|x, u) is continuous in total vari-
+ation in (x, u), i.e., for any (xn, un) → (x, u), T (·|xn, un) → T (·|x, u) in total varia-
+tion.
+Theorem 1.1.
+(i) [22] Under Assumption 1, the transition kernel η(F(π, u, Y1) ∈ ·|π, u) of the
+filter process is weakly continuous in (π, u).
+(ii) [35] Under Assumption 2, the transition kernel η(F(π, u, Y1) ∈ ·|π, u) of the
+filter process is weakly continuous in (π, u).
+If the cost function c(x, u) is continuous and bounded, an application of the
+dominated convergence theorem implies that ˜c(z, u) = Ez[c(X, u)] is also continuous
+and bounded.
+In the uncontrolled setting, [5] and [15] have established similar weak continuity
+conditions (i.e., the weak-Feller property) of the non-linear filter process (i.e., the
+belief process) in continuous time and discrete time, respectively, where Assumption
+1 is present for an additive noise measurement model.
+4
+
+The convex analytic approach [45, 11] is a powerful approach to the optimiza-
+tion of infinite-horizon problems. It is particularly effective in proving results on the
+optimality of stationary (and possibly randomized stationary) policies, through an
+infinite-dimensional linear program for constrained optimization problems and infi-
+nite horizon average cost optimization problems. For the average cost criterion, recall
+that via (1.6)-(1.7) under the belief-MDP reduction, we are interested in the mini-
+mization:
+inf
+γ∈Γ lim sup
+N→∞
+1
+N Eγ
+µ[
+N−1
+�
+t=0
+˜c(πt, ut)],
+(1.8)
+where Eγ
+x0[·] denotes the expectation over all sample paths with initial state given by
+x0 under the admissible policy γ.
+Assumption 3.
+(i) The R+-valued one-stage cost function c is bounded and continuous.
+(ii) Assumption 1 or 2 holds (this leads to the belief-MDP η to be weakly contin-
+uous by Theorem 1.1).
+(iii) X and U are compact.
+Theorem 1.2. Under Assumption 3, there exists an optimal invariant measure
+and an associated optimal control policy, for either average cost (1.4) or for the dis-
+counted cost (1.5) criteria.
+Proof. For the average cost criterion, the proof follows from the convex analytic
+method, studied by Borkar in the weakly continuous case [11]; for completeness, and
+its use later in the paper, a proof is presented in Section A.1. For the discounted
+cost criterion, the result follows from a standard verification and measurable selection
+analysis on the discounted cost optimality equation [30].
+While this result presents sufficient findings on the existence of optimal control
+policies, the approach has a number of limitations.
+(i) [Fully observed MDPs viewed as a special case of POMDPs] Consider the
+case with the state transition kernel T being weakly continuous and where
+the measurements satisfy yt = xt, that is, with full state information, in which
+case the measurement kernel is Q(dy|x) = δx(dy), and thus the system does
+not satisfy Assumption 1 or 2 (as the channel is only weakly continuous but
+not total variation continuous). On the other hand, this model, which is a
+fully observed setup, has been studied well and leads to well-known existence
+(and further, structural and approximation) results. Therefore, we observe
+that one should be able to relax the conditions further, which we will indeed
+find to be the case.
+(ii) For the average cost criterion, Assumption 3 provides weaker conditions when
+compared with the efforts in the literature that typically have utilized the van-
+ishing discount approach: [10] considers the finite model setup and [60] con-
+sider restrictive assumptions on the controlled state process, and [21] which in
+the POMDP case would require yet to be established rates of convergence con-
+ditions to invariant measures (needed for uniform boundedness properties of
+relative discounted cost value functions as the discount parameter approaches
+unity). We will see that further relaxations can be provided.
+(iii) There is a subtle question of how to select the initial measure, that is the
+question of achieving the optimal cost under a given initial belief measure
+and whether the support of an invariant probability measure attracts the be-
+lief process from a given initial condition. For an interesting counterexample,
+5
+
+see [37, Example 2.3]. For belief MDPs, there are few results on unique ergod-
+icity properties, therefore the impact of the initial priors is an open problem
+under the belief-separated-approach. In our case, the infinite dimensional
+formulation to be presented leads to a flexibility on how to select the initial
+prior which we show to lead to global optimality under a mild and testable
+absolute continuity condition (this approach does not seem to easily carry
+over to the belief-separated-approach).
+(iv) Related to (iii), the unique ergodicity problem for the controlled or uncon-
+trolled non-linear filtering process entails open problems. An alternative ap-
+proach to what currently exists in the literature is needed, especially for the
+controlled setup. For the control-free case, building on [46, Theorem 2] and
+[61, Prop 2.1], it can be shown that almost sure filter stability in the total
+variation sense (or weak merging sense, with the arguments tailored to this
+case) and the uniqueness of an invariant probability measure on the hidden
+state process leads to unique ergodicity of the filter process. We will see that a
+complementary condition for the control-free case can be established (unique
+ergodicity of the measurement process and a measurability condition related
+to filter stability also leads to unique ergodicity).
+1.2. Statement of main results and contributions. The paper makes the
+following contributions.
+(i) We present an approach of defining the state as an infinite-dimensional con-
+trolled Markov chain where the control policies satisfy conditional indepen-
+dence between the state and control actions given the information. For the
+controlled case, we provide new sufficient conditions for the existence of opti-
+mal control policies which turn out to be stationary in the history variables.
+In particular, while in the belief-MDP reduction of POMDPs, weak Feller
+condition requirement imposes total variation continuity on either the sys-
+tem kernel or the measurement kernel, with the approach of this paper only
+weak continuity of both the transition kernel and the measurement kernel is
+sufficient together with a stability condition (Theorem 3.3). For the aver-
+age cost criterion, the paper provides a general approach on how to generate
+initial priors and beliefs so that optimal performance can be attained. The
+infinite dimensional formulation presents a natural flexibility in initialization,
+which can be used to arrive at optimal performance under a mild absolute
+continuity condition, see Theorem 3.5 (which does not seem to be lenient
+under the standard belief-MDP approach).
+(ii) For the discounted cost criterion, more relaxed existence conditions will be
+presented in Section 4. We first establish the existence of a stationary optimal
+policy. We then establish near optimality of finite window policies via a direct
+argument involving near optimality of quantized approximations for MDPs
+under weak Feller continuity (see Theorem 4.1), where finite truncations of
+memory can be viewed as (uniform) quantizations of infinite memory (which is
+the state definition adopted for the discounted cost criterion by Theorem 2.5)
+with a uniform diameter in each finite window restriction under the product
+metric. Building on recent results on near-optimality of quantized policies
+for weak Feller MDPs [55], near optimality of finite memory policies follows,
+complementing and generalizing recent works on the subject [39, 40]. This
+also facilitates reinforcement learning theoretic methods for POMDPs, which
+is discussed in further detail in the paper.
+6
+
+(iii) Via our approach, further results on existence and uniqueness of invariant
+probability measures for control-free non-linear filter processes will be estab-
+lished in Section 5. In particular, we will see that unique ergodicity of the
+measurement process and a measurability condition related to filter stability
+leads to unique ergodicity (this complements, with an alternative argument,
+several results in the literature).
+2. Another Look at POMDPs: Infinite Dimensional Markov Decision
+Process Formulation.
+2.1. Controlled-Markov construction with infinite-memory state space
+for the average cost criterion. To facilitate the solution approach presented in
+this paper, we first assume that the measurements and the control actions have been
+taking place since −∞. Later on we will motivate and justify this approach.
+Let YZ− be the one-sided product space consisting of elements of the form
+y = {· · · , yn, · · · , y−2, y−1, y0}
+with yk ∈ Y. We endow YZ− with the product topology; this makes YZ− a metric
+space, which is complete and separable. Likewise, we will view U(−∞,−1] also as a
+UZ−-valued random variable.
+A related view, on using the infinite past as a Markov process, was presented in
+[66] to establish stochastic stability of control-free dynamical systems driven by noise
+processes which are not necessarily independent and identically distributed, but which
+are stationary. A further related approach was adopted in [1]; see also [28] (as well as
+[7] and [67] as discussed earlier).
+Theorem 2.1. With
+Zk = (Y(−∞,k], U(−∞,k−1], Xk),
+the pair (Zk, Uk) is a controlled Markov chain where Zk is YZ− × UZ− × X valued.
+Proof. Let A = �0
+k=−∞ Ay
+k be an open cylinder set in YZ− and Au = �0
+k=−∞ Au
+k
+be an open cylinder set in UZ− and let B ∈ X be a Borel set:
+P
+�
+Zt+1 ∈ (Ay × Au × B)|Zt = z, Ut = u, Z[0,t−1] = z[0,t−1], U[0,t−1] = u[0,t−1]
+�
+=
+�
+Q(yt+1 ∈ Ay
+0|xt+1)T (xt+1 ∈ B|xt = x, ut = u)
+×P(y(−∞,t] ∈
+0
+�
+k=−∞
+Ay
+k|y(−∞,t])P(u(−∞,t] ∈
+0
+�
+k=−∞
+Au
+k|u(−∞,t])
+=
+�
+Q(yt+1 ∈ Ay
+0|xt+1)T (xt+1 ∈ B|xt = x, ut = u)1{y(−∞,t]∈�0
+k=−∞ Ay
+k,u(−∞,t]∈�0
+k=−∞ Au
+k}
+(2.1)
+=: P(Ay × Au × B|Zt = z, Ut = u)
+(2.2)
+where for a Borel C, Q(y ∈ C|x) := P(G(X0, V0) ∈ C|X0 = x) by (1.2) and T is the
+transition kernel defined by (1.1).
+In the above, we define P to be the transition kernel for this Markov chain.
+Definition 2.2 (Weak-Feller Condition for P). P(dzt+1|zt, ut) is weak-Feller if
+for every continuous and bounded f ∈ Cb(YZ− × UZ− × X):
+�
+f(zt+1)P(dzt+1|zt = z, ut = u)
+7
+
+is continuous in (z, u).
+The following will be invoked often.
+Assumption 4. Q(·|x) is weakly continuous in x, and T (·|x, u) is weakly con-
+tinuous in x, u.
+Lemma 2.3. Under Assumption 4, P is weak-Feller.
+Proof. First note that by (2.1) we have
+P(dzt+1|zt, ut) = Q(dyt+1|xt+1)T (dxt+1|xt, ut)P(dy(−∞,t]|y(−∞,t])P(du(−∞,t]|u(−∞,t])
+where P(y(−∞,t] ∈ ·|y(−∞,t]) = δy(−∞,t](·) is the Dirac measure (as in (2.1)), with the
+same holding for P(du(−∞,t]|u(−∞,t]).
+We wish to show that the following is continuous in xt, y(−∞,t], u(−∞,t−1], ut, for
+continuous and bounded f:
+�
+f
+�
+xt+1, y(−∞,t+1], u(−∞,t]
+�
+P(dxt+1, dy(−∞,t]+1, du(−∞,t−1|xt, y(−∞,t], u(−∞,t−1, ut)
+=
+�
+f
+�
+xt+1, y(−∞,t+1], u(−∞,t]
+�
+Q(dyt+1|xt+1)T (dxt+1|xt, ut)
+×P(dy(−∞,t]|y(−∞,t])P(du(−∞,t]|u(−∞,t])
+=
+� �
+xt+1
+�� �
+yt+1
+f(xt+1, y(−∞,t+1], u(−∞,t])Q(dyt+1|xt+1)
+�
+T (dxt+1|xt, ut)
+�
+×P(dy(−∞,t]|y(−∞,t])P(du(−∞,t]|u(−∞,t])
+The expression
+g(xt+1, y(−∞,t+1], u(−∞,t) :=
+� �
+yt+1
+f(xt+1, y(−∞,t+1], u(−∞,t)Q(dyt+1|xt+1)
+�
+is continuous by a generalized dominated convergence theorem for varying measures
+under continuous-convergence [44, Thm.
+3.5] (see also [57, Thm. 3.3] for a more
+restricted version). Since g is continuous in xt+1,
+h(xt, ut, y(−∞,t], u(−∞,t]) :=
+�
+xt+1
+�
+g(xt+1, y(−∞,t], u(−∞,t])T (dxt+1|xt, ut)
+�
+is also continuous in its parameters. Finally, again by [44, Thm. 3.5],
+�
+h(xt, ut, y(−∞,t], u(−∞,t])P(dy(−∞,t]|y(−∞,t])P(du(−∞,t]|u(−∞,t])
+is continuous in xt, y(−∞,t], u(−∞,t−1], ut.
+The above completes the description of the controlled Markov model, that we will
+utilize for the average cost problem.
+2.2. Key technical assumptions on continuity and stability. We will oc-
+casionally invoke a subset of the following assumptions on stationarity, stability, and
+continuity. We will see, in Appendix B, that these are related to filter stability [17].
+Assumption 5. [A Continuity Condition] The stochastic kernel P(dxk|y(−∞,k], u(−∞,k−1])
+is weakly continuous, that is, for every continuous and bounded g : X → R,
+�
+g(x)P(Xk ∈ dx|y(−∞,k], u(−∞,k−1])
+8
+
+is continuous in (y(−∞,k], u(−∞,k−1]) (under the product topology on YZ− × UZ−).
+It will be shown in the appendix that a uniform filter stability condition, such
+as [27, Corollary 4.2] (via a Hilbert metric approach) as well as [47, Corollary 3.7]
+(via Dobrushin’s coefficient method), implies Assumption 5 for the case where the
+measurement and action spaces are finite. Further discussion is available in Appendix
+B. Our next assumption is the following.
+Assumption 6. [A Stationarity / Stability Condition] For every Borel B, and
+with y = (· · · , y−2, y−1, y0), u = (· · · , u−2, u−1, u0), under any control policy, for
+almost every (y, u), we have that
+P(Xt ∈ B|Y(−∞,t] = y, U(−∞,t−1] = u) = P(Xt+1 ∈ B|Y(−∞,t+1] = y, U(−∞,t] = u)
+This assumption is also implied by an almost sure filter stability condition. In fact,
+[27, Corollary 4.2] also implies the next, and final, assumption: P
+�
+Xt ∈ ·
+����
+�∞
+n=1 σ(Y(−∞,t], U(−∞,t−1])∨
+σ(π(−∞,−n])
+�
+is σ(It)-measurable in the sense that P a.s.
+P
+�
+Xt ∈ ·
+����
+∞
+�
+n=1
+σ(Y(−∞,t], U(−∞,t−1]) ∨ σ(π(−∞,−n])
+�
+= P(Xt ∈ ·|σ(Y(−∞,t], U(−∞,t−1]))
+(2.3)
+Let πt be the conditional probability on the state, given a (distant) prior at time
+−n ∈ Z−, and the information since then (or, equivalently, the information even prior
+to −n): For all Borel A ⊂ X
+πt(A) = E[1Xt∈A|σ(Y(−∞,t], U(−∞,t−1]) ∨ σ(π(−∞,−n]]
+In view of the above, we state the following assumption.
+Assumption 7. [A Filter Stability Condition] πt is σ(It) measurable where It =
+(Y(−∞,t], U(−∞,t−1]). In particular, there exists a measurable g such that g(y(−∞,t], u(−∞,t−1])(A) =
+πt(A) for every Borel A.
+Lemma 2.4. Condition (2.3) implies Assumption 7.
+Proof. Observe that πt(A) = E[1Xt∈A|σ(Y(−∞,t], U(−∞,t−1]) ∨ σ(π(−∞,−n]] for
+every n > 0 and thus
+πt(A) = lim
+n→∞ E[1Xt∈A|σ(Y(−∞,t], U(−∞,t−1]) ∨ σ(π(−∞,−n]].
+Now, E[1Xt∈A|σ(Y(−∞,t], U(−∞,t−1])∨σ(π(−∞,−n]] is a bounded backwards martingale
+sequence with respect to the decreasing filtration σ(Y(−∞,t], U(−∞,t−1])∨σ(π(−∞,−n]),
+n ∈ N and by the backwards martingale theorem [19], the above limit will converge
+to
+P
+�
+Xt ∈ A
+����
+∞
+�
+n=1
+σ(Y(−∞,t], U(−∞,t−1]) ∨ σ(π(−∞,−n])
+�
+This then implies, via (2.3), πt(A) is σ(It)-measurable for any given A.
+Recall the following which builds on Theorem 2.1 of Dubins and Freedman [18] and
+Proposition 7.25 in Bertsekas and Shreve [4]: Let S be a Polish space, P(S) be the set of
+9
+
+probability measures under the weak convergence topology and (M, M) be a measurable
+space.
+A function F : (M, M) → P(S) is measurable on M if for all B ∈ B(S)
+(F(·))(B) : M → R is measurable on M, that is for every B ∈ B(S), (F(π))(B) is a
+measurable function when viewed as a function from M to R. See also [4, Proposition
+7.26]. By this result, it follows that πt is measurable on �∞
+n=1 σ(Y(−∞,t], U(−∞,t−1]) ∨
+σ(π(−∞,−n]). But by condition (2.3), it is also σ(It) measurable, and as the considered
+spaces are standard Borel, a functional representation via the measurable function g
+follows.
+Further analysis and sufficient conditions for Assumptions 5, 6 and 7 are presented
+in Appendix B. We note that (2.3) is essentially a filter stability condition: Indeed,
+the assumption above, in the control-free setup, is related to the statement in [16,
+Theorem 3.1(2)], building on [43].
+2.3. Controlled-Markov construction with infinite-memory state space
+for the discounted cost criterion. Given the construction and stability properties,
+for the discounted cost criterion, towards arriving at approximate optimality results
+using finite window policies, we will present an alternative controlled Markov model.
+The proof follows from identical arguments presented in Section 2.1, but here we
+apriori impose Assumption 5 and Assumption 7.
+Theorem 2.5.
+(i) With
+Sk = (Y(−∞,k], U(−∞,k−1]),
+the pair (Sk, Uk) is a controlled Markov chain where Sk is YZ− × UZ− valued.
+(ii) Under Assumption 7, we have that for all u ∈ U
+E
+�
+c(Xk, u)
+����(Y(−∞,k], U(−∞,k−1] = s)
+�
+= ¯c(s, u),
+for some measurable ¯c : YZ− × UZ− × U → R.
+(iii) Under Assumption 5, ¯c is continuous and bounded, when c is bounded con-
+tinuous.
+(iv) The kernel P(zt+1 ∈ ·|Zt = z, Ut = u) is weak Feller under Assumptions 4
+and 5.
+The final result, (iv), in the above follows by considering P(Yt ∈ dy|Zt = z, Ut =
+u) =
+�
+Q(dy|x)P(Xt ∈ dx|Zt = z, Ut = u), and the weak continuity of each of the
+kernels in the integration and generalized weak convergence [44, Thm. 3.5] as used
+earlier in the proof of Lemma 2.3; (iii) follows from Assumption 5 by definition. (i)
+follows by construction, and (ii) directly by Assumption 7.
+As a result, under Assumptions 7 and 5, we have an equivalent controlled Markov
+model with cost ¯c : (YZ− ×UZ−)×U → R+, YZ− ×UZ−-valued state sk, and transition
+kernel P(Zt+1 ∈ ��|Zt = z, Ut = u).
+2.4. Conditionally state independent control policies. Once we have es-
+tablished controlled Markov models, we now discuss the classes of control policies
+considered under the infinite dimensional controlled Markov model presented. We
+recall that in the theory of stochastic control, to facilitate stochastic analysis for arriv-
+ing at existence, structural or approximation results (e.g. via continuity-compactness
+properties), the set of control policies may be enlarged. This is often referred to as a
+relaxation of control policies. Relaxations have been very effective in optimal control,
+10
+
+with a very prominent example being Young measures in deterministic optimal control
+[64]. These allow one to use topologies on the sets of probability measures to study
+existence, optimality, and structural results (especially that of weak convergence),
+rather than working with the space of measurable functions only (whose compactness
+conditions under an appropriate metric would be too restrictive). A key aspect of such
+relaxations is that any relaxation should not allow for optimal expected cost values
+to be improved; they should only be means to facilitate stochastic analysis. We can
+classify some policies and various relaxations as follows:
+(i) Wide sense admissible policies introduced by Fleming and Pardoux [25] and
+prominently used to establish the existence of optimal solutions for partially observed
+stochastic control problems. Borkar [10, 13, 12] (see also Borkar and Budhiraja [14])
+have utilized these policies for a coupling/simulation method to arrive at optimality
+results for average cost partially observed stochastic control problems. The approach
+here is to first apply a Girsanov type (see Borkar [10, 13] for discrete-time models and
+Witsenhausen [62] for decentralized stochastic control where the change of measure
+argument leads to static reduction) transformation to decouple the measurements from
+the system via an absolute continuity condition of measurement variables conditioned
+on the state, with respect to some reference measure; and then use independence
+properties: In the discrete-time case, {Yn} is i.i.d. and independent of X0 and the
+system noise {Wn}, and {U0, . . . , Un, Y0, . . . , Yn} is independent of {Wn}, X0, and
+{Ym, m > n}, for all n.
+(ii) Policies defined by conditional independence (as in the dynamic programming
+formulation [67] for decentralized stochastic control).
+In the context of our setup
+here, the actions are those that satisfy Un being conditionally independent of the past
+(X0, W[0,n], V[0,n]) given the local information {Y[0,n], U[0,n−1]}:
+Un ↔ {Y[0,n], U[0,n−1]} ↔ {X0, W[0,··· ), V[0,··· )}
+(2.4)
+We note that, here an absolute continuity condition required for the static reduction
+or Girsanov-type measure transformation is not necessary apriori. We will call such
+policies Conditionally-Exogenous Variable-Independent Policies since the conditional
+independence holds between action, information, and the exogenous random variables
+in the system.
+In our paper, we will consider a further refinement (to be called
+Conditionally-State-Independent Policies) which satisfies instead of (2.4)
+Un ↔ {Y[0,n], U[0,n−1]} ↔ Xn
+(2.5)
+to facilitate the infinite horizon analysis.
+If we have independent static reduction, with measurements ˜Yn, by considering
+the method for static measurements with expanding information structure (this is
+the information structure that one would obtain for a POMDP under the absolute
+continuity conditions) in [68, Theorem 5.6] or [54, Theorem 4.7] by expressing all the
+cost-relevant uncertainty in terms of ˜Yn, we have the following version of (2.4)
+Un ↔ { ˜Y[0,n], U[0,n−1]} ↔ { ˜Y[n+1,··· )}
+(2.6)
+If the process continues on since the indefinite past, we write (2.5) with
+Un ↔ {Y(−∞,n], U(−∞,n−1]} ↔ Xn
+(2.7)
+3. Existence Results for Optimal Policies: Average Cost Criterion.
+11
+
+3.1. On the existence of an optimal stationary policy. We first present a
+supporting result in the following, which will be crucial in our analysis to follow.
+Lemma 3.1. a) Let Y 1
+n , Y 2
+n , U 2
+n be a sequence of random variables that satisfies
+for every n, the conditional independence property:
+Y 1
+n ↔ Y 2
+n ↔ U 2
+n,
+with joint measure Pn, where the marginal on (Y 1
+n , Y 2
+n ) is fixed throughout the se-
+quence. In this case, if Pn → P weakly, the limit measure P also satisfies
+Y 1 ↔ Y 2 ↔ U 2
+b) Let Y 1
+n , Y 2
+n , U 2
+n be a sequence of random variables that satisfies for every n:
+Y 1
+n ↔ Y 2
+n ↔ U 2
+n
+with measure Pn where the conditional probability Pn(Y 1
+n ∈ dy1|Y 2
+n = y2) = κ(dy1|y2)
+is fixed throughout the sequence and that κ(dy1|y2) is a weakly continuous kernel (i.e.
+y2 �→
+�
+f(y1)κ(dy1|y2) is continuous and bounded for every bounded continuous f).
+In this case, if Pn → P weakly, then, the limit measure also satisfies
+Y 1 ↔ Y 2 ↔ U 2
+Proof. See Section A.2.
+We now present a key intermediate result.
+Lemma 3.2. Let X, Y, U be compact. Under Assumptions 4, 6 and 5, the set of
+invariant occupation measures
+G = {v ∈ P(X × YZ− × UZ−) :
+v(B × U) =
+�
+z,u
+P(zt+1 ∈ B|z, u)v(dz, du),
+B ∈ B(X × YZ− × UZ−)}
+that simultaneously satisfy the Markov chain
+X0 ↔ (U(−∞,−1]), Y(−∞,0]) ↔ U0,
+or, equivalently, that belong to
+H = {v ∈ P(X × YZ− × UZ−) :
+v(dx0, dy(−∞,0], du(−∞,−1], du0)
+= v(dx0, dy(−∞,0], du(−∞,−1])P(du0|dy(−∞,0], du(−∞,−1])},
+(3.1)
+is a weakly compact set.
+Proof. We note that under weak continuity of the transition kernel for P, G is a
+closed set under weak convergence. By Lemma 3.1(b), as the kernel
+P(dxt|y(−∞,t], u(−∞,t−1])
+is weakly continuous (by Assumption 5) and time-invariant (by Assumption 6) , H
+is also closed. Since the state space is compact, the set of probability measures on
+(X×YZ− ×U) is tight. The result follows since a closed subset of a tight set is weakly
+compact.
+12
+
+Building on the above, we are able to state the following:
+Theorem 3.3. Let X, Y, U be compact and Assumptions 4, 6 and 5 hold. Then
+an optimal control policy exists. This policy is stationary and can be written as:
+Ut = γ(Y(−∞,t], U(−∞,t−1], Rt),
+where Rt is an i.i.d. sequence.
+Proof. See Section A.3.
+Remark 3.1. The above is a relaxation on the known results where one needs
+to reduce the POMDP to a belief-MDP which is weak Feller. As noted earlier, the
+weak Feller conditions require total variation continuity on either the system kernel
+or the measurement kernel. Only weak continuity of both the transition kernel and
+the measurement kernel is needed for the result above, though filter stability is also
+imposed.
+3.2. On the initial state distribution realization for an optimal station-
+ary policy. In the above, there is an important operational question: The actual
+system is a one-sided process with X0 ∼ µ being the starting variable. To address
+this issue, in the following we explain that we can randomly generate the past and use
+the past realizations to apply optimal control. Under some conditions, the average
+cost will converge to the optimal cost.
+Our strategy on the initialization is as follows: (i) We will first show that an
+optimal invariant measure can be taken to be ergodic (i.e., an invariant measure
+which cannot be expressed as a convex combination of multiple invariant measures).
+(ii) Then we will construct an initialization which is in the attractor set of this optimal
+invariant measure, under a mild absolute continuity condition.
+We will generate
+the measurement and control actions Y(−∞,−1], U(−∞,−1] according to a stationary
+measure, and then X0 ∼ µ, Y0 accordingly, and continue with the process according
+to the optimal policy under Theorem 3.3.
+Lemma 3.4.
+Without any loss of optimality, an optimal invariant occupation
+measure can be assumed to be ergodic.
+Proof. Let ν be an optimal occupation measure, which leads to a policy γ. Under
+γ, the state process
+(Y(−∞,k], U(−∞,k−1], Xk),
+is a Markov chain with the marginal of the invariant measure ν on this state being
+invariant. By an ergodic decomposition theorem, every stationary measure can be
+expressed as a convex combination of ergodic invariant measures with disjoint sup-
+ports: let ν be expressed as a convex combination of measures νβ, parametrized by β
+(see Theorem C.1 for some related discussion), where each of them are also invariant
+and ergodic. Now, a question is whether each of the νβ measures satisfies (3.1) and
+(3.1). The first holds by the invariance of each of these measures. The latter holds
+by definition: conditioned on the support of each νβ, ν satisfies (3.1) (note that the
+control policy is fixed) and hence this holds also for νβ.
+Since the cost in (A.14):
+inf
+v∈G∩H
+�
+v(dx, dy(−∞,0], du(−∞,−1], du)c(x, u)
+(3.2)
+is linear in the space of (signed) measures, we conclude that without any loss we can
+take ν to be from the set of ergodic invariant measures.
+13
+
+The above then completes our realization result:
+Theorem 3.5. Let Assumptions 5 and 6 hold. Let Ut = γ(Y(−∞,t], U(−∞,t−1], Rt)
+be an optimal stationary policy with Rt being an i.i.d. noise process. Let
+g(y(−∞,k], u(−∞,k−1], xk) := E[c(xk, γ(y(−∞,k], u(−∞,k−1], Rk))]
+where the expectation is over Rk. Let ¯P denote an (invariant) process measure on
+(Y(−∞,k−1], U(−∞,k−1], Yk, Xk),
+and let P0 be the marginal of the invariant measure on {Y(−∞,−1], U(−∞,−1]} and µ
+be the prior measure on X0 as imposed (i.e., given in the problem statement) on the
+controller/decision maker. If P0 × µ ≪ ¯P (note that the distribution on X0 specifies
+that on Y0); that is, absolute continuity holds when the (past history) initialization is
+independent of the state process initialization, then
+lim
+T →∞
+1
+T EP0×µ
+� T −1
+�
+k=0
+c(Xk, γ(Y(−∞,k], U(−∞,k−1], Rk))
+�
+=
+�
+g(y(−∞,0], u(−∞,−1], x0) ¯Q(dy(−∞,0], du(−∞,−1], x0),
+(3.3)
+for some invariant ¯Q, which would necessarily be ¯P under the assumed ergodicity of
+¯P via Lemma 3.4, in which case the optimal control/decision cost will be attained.
+Remark 3.2. Note that, in the above, with ¯P and P0 both stationary, we have
+that ¯P decomposes as P0(dy(−∞,−1], du(−∞,−1]) ¯P(dx0|y(−∞,−1], u(−∞,−1]). A suffi-
+cient condition for the aforementioned absolute continuity condition then is, ¯P a.e.,
+µ(dx0) ≪ ¯P(dx0|dy(−∞,−1], du(−∞,−1])
+Remark 3.3. Observe that in the analysis above we constructed y(−∞,−1], u(−∞,−1]
+and not y(−∞,0], since µ on X0 induces directly a measure on Y0. Note that with P0×µ
+viewed as an initial measure, when the process evolves, the evolution is consistent with
+the actual realizations under the true measure. That is, the initialization does not af-
+fect the evolution of the measure. The reason is that the process X0 determines the
+probability of the future events, independent of the history initialization via:
+P(dzt+1|zt, ut) = Q(dyt+1|xt+1)P(dxt+1|xt, ut)P(dy(−∞,t]|y(−∞,t])P(du(−∞,t]|u(−∞,t]).
+4. Discounted Cost Criterion:
+Refined Existence Results and Near
+Optimality of Finite Window/Memory Policies. In this section, we study the
+discounted cost setup. We will see a particular utility in this criterion: near optimality
+of finite window policies.
+For the discounted criterion, we will restrict our control policies to those that are
+strict-sense admissible. We could also consider the relaxed framework, however the
+analysis for the discounted criterion setup will be seen to be more direct. Compared
+with the average cost criterion, we are able to utilize a verification theorem directly
+following the discounted cost optimality equation (DCOE).
+14
+
+Let, as in Theorem 2.5, sk = (y(−∞,k], ˜u(−∞,k−1]) ∈ YZ− × UZ−. Now, under
+strict-sense policies, with the assumptions on weak-Feller continuity and bounded
+continuous cost function, if there exists a solution to
+Jβ(s) = min
+uk∈U E
+�
+c(xk, uk) + βJk+1(Sk+1)|Sk = s, Uk = u
+�
+,
+then we can declare this policy to be optimal using a standard verification argument
+[30]. As before, we assume that X, Y, U are compact.
+Theorem 4.1. An optimal solution exists under Assumptions 4 and 5 and As-
+sumption 7.
+Proof. Note first that
+E[(c(Xk, Uk))|Sk = s, Uk = u]
+will be a measurable function of (s, u) under Assumption 7, call this ¯c(s, u). We then
+have
+Jβ(s) = min
+u∈U
+�
+E[¯c(Sk, Uk) + βJk+1(Sk+1)|Sk = s, Uk = u]
+�
+Then, if we have weak-Feller continuity of the controlled kernel (by Assumption 4) and
+if we have the continuity of f (by Assumption 5), by measurable selection theorems
+[30], there exists a solution to the discounted cost optimality equation.
+As a consequential application, we note the following. We say that, for some
+N ∈ N, a control policy is an N-memory policy if for t > N,
+Ut = γt(Y[t−N+1,t], U[t−N+1,t−1]),
+and for 0 ≤ t ≤ N,
+Ut = γt(Y[0,t], U[0,t−1]).
+Theorem 4.2. Under the conditions of Theorem 4.1, the set of N-memory poli-
+cies are asymptotically optimal; i.e., for every ǫ > 0, there exists N so that N window
+policies are ǫ-optimal.
+Before the prove the theorem, let us note that the near-optimality is operationally
+justified in two scenarios. If the costs start becoming active from time N onwards, or
+the policies in the first N stages are designed via a policy-iteration argument; see [39,
+p. 16].
+Proof. Finite window truncation of z can be viewed as a uniform binning/quantization
+(even though the number of bins may not be countable) operation applied to the
+infinite-dimensional state vector z under the product topology. Indeed, define (the
+binning/quantization map)
+ρ : YZ− × UZ− → YN × UN−1,
+with
+ρ(y(−∞,0], ˜u(−∞,−1]) = (y[−N,0], u[−N,−1])
+In this case, for any (¯y(−∞,0], ¯u(−∞,−1]) and (˜y(−∞,0], ˜u(−∞,−1]) with
+ρ(¯y(−∞,0], ¯u(−∞,−1]) = ρ(˜y(−∞,0], ˜u(−∞,−1]),
+15
+
+and ¯d being the product metric defined by
+¯d(¯y(−∞,0], ¯u(−∞,−1], ˜y(−∞,0], ˜u(−∞,−1])
+:=
+∞
+�
+k=0
+2−k
+dY(¯yk, ˜yk)
+1 + dY(¯yk, ˜yk) +
+∞
+�
+m=0
+2−m
+dU(¯um, ˜um)
+1 + dU(¯um, ˜um),
+(4.1)
+where dY and dU are the metrics on Y and U, respectively; we have that
+¯d(¯y(−∞,0], ¯u(−∞,−1], ˜y(−∞,0], ˜u(−∞,−1]) ≤ 2−N+1.
+Since the constructed kernel P is weakly continuous, the cost function is bounded
+continuous, and the state space is compact, the proof is then a corollary of [53,
+Theorem 4.27] (see also [41, Theorem 11]) by viewing the finite window truncation
+as a quantization (with a uniformly bounded radius for each bin) of the state space
+under the product topology.
+In particular, one can construct an approximate MDP model with state space
+YN × UN−1 and action space U and transition kernel PN whose solution can be
+extended, via the quantization rule ρ, to the whole YZ− × UZ− and which will be near
+optimal for the original problem [53, Theorem 4.27].
+Algorithmic and Numerical Implications. Note that a finite memory trun-
+cation leads to a uniform quantization error, therefore the mathematical analysis
+adopted in this paper is particularly suitable for such a problem in view of recent
+near optimality results for quantized approximations to weakly continuous kernels.
+We note that Theorem 4.2 has a rather direct relation with [39] and [40], where near
+optimality of finite window policies were established via alternative and more tedious
+methods, and more restrictive conditions (though with rates of convergence prop-
+erties, which we do not present). In our current paper, the filter stability condition
+manifests itself via Assumption 7 with no additional assumptions other than the weak
+Feller property. The result above thus presents further sufficient conditions on the
+applicability of re-inforcement learning methods presented in [40] for finite memory
+near-optimal control. In particular, if one applies an independently randomized ex-
+ploration control policy for ut (for learning) in the Q-learning algorithm presented in
+[40, Section 4], one arrives at a control-free system under which the unique ergodicity
+of the measurement process is satisfied under mild conditions so that the conditions in
+[40, Theorem 4.1] (and in this particular fully observed interpretation, [36, Corollary
+3.3]) is applicable for arriving at near-optimal control policies which use only a finite
+window of recent measurement and past actions.
+We also note here that another positive attribute for the discounted criterion is
+that under weak Feller continuity, both the value functions and optimal policies are
+robust to model approximations [38, Theorem 4.4].
+5. The Control-Free Case: Implications on Unique Ergodicity of Non-
+Linear Filters and Existence of an Optimal Stationary Policy. In this section,
+we interpret some of the results presented in the controlled-case in the context of the
+control-free case. This serves both as an application and validation of the approach,
+but also presents new results on the non-linear filtering problem.
+Even though the control-free case can be viewed as an instance of the controlled
+case, here some of the assumptions can be relaxed. Accordingly, we will state two
+results. One is on the stationarity properties on the optimal decision policies, and the
+other is on how to realize it by selecting the initial prior appropriately.
+16
+
+We have the following assumption for some of the results to follow. This will be
+a complementary condition.
+Assumption 8. {Xt} is stationary.
+Note that if Xt is stationary, so is the pair process (Xt, Yt).
+By a standard
+argument (e.g. Chapter 7 in [19]), we can embed the one-sided stationary process
+{Xk, k ∈ Z+} into a bilateral (double-sided) stationary process {Xk, k ∈ Z}.
+Theorem 5.1.
+Let X, Y, U be compact and Assumption 4 hold. Under either
+Assumption 8 or Assumptions 5 and 6 an optimal decision/control policy exists. This
+policy is stationary.
+Proof. See Section A.5.
+Corollary 5.2.
+Let either (i) Assumption 8, or, (ii) under the hypotheses
+in Theorem 5.1, Assumption 7, hold. Then, the filter process admits an invariant
+probability measure.
+As noted above, in the uncontrolled setting, [5] and [15] have established weak
+continuity conditions (i.e., the weak-Feller property) of the non-linear filter process
+(i.e., the belief process) in continuous time and discrete time, respectively; where the
+total variation continuity of the measurement channel was imposed. These results
+were used to establish the existence of an invariant probability measure for the belief
+process. The above shows that for the existence of an invariant probability measure,
+one may not need to invoke the continuity conditions.
+A corollary (to Theorem 5.1 and Lemma 2.4) is the following.
+Corollary 5.3. If Yt is uniquely ergodic and if Assumption 7 holds, then the
+filter is uniquely ergodic.
+Proof. We have seen that there exists at least one invariant probability measure.
+Suppose that there were two distinct invariant probability measures, η1 and η2. Since a
+countable collection of continuous and bounded functions can be used to distinguish
+probability measures, we will consider
+�
+ηi(dπ)f(π) for such a countable collection
+of functions f, for i = 1, 2. Consider the joint process (πt, Y(−∞,t]) with invariant
+measure κi and with a marginal invariant measure on πt as ηi. Since the marginal on
+Y(−∞,t] is uniquely ergodic, for every invariant measure κi on the joint process, the
+marginal will be a constant measure, call ψ. Therefore:
+�
+ηi(dπ)f(π) =
+�
+κi(dπ, dy(−∞,t])f(π) = Eκi[f(π)] = Eκi[Eκi[f(π)|Y(−∞,t]]]
+However, Eκi[f(π)|Y(−∞,t]] = Eκi[g(Y(−∞,t])] = Eψ[g(Y(−∞,t])] for some measurable
+g, by Assumption 7. Therefore, as this argument applies for any f from the distin-
+guishing family, we must have that η1 = η2.
+Note that for Yt to be uniquely ergodic, we don’t require Xt to be uniquely
+ergodic. On the other hand, if Xt is uniquely ergodic, Yt must be uniquely ergodic as
+well. For sufficient conditions on unique ergodicity of infinite-memory processes such
+as Yt, see [66, Theorem 3.8]; in particular the existence of an accessible element and a
+continuity condition (instead of weak continuity, where setwise continuity is imposed
+in Assumption 5; e.g., [27, Corollary 4.2] implies this condition as well).
+Notably, building on [46, Theorem 2] and [61, Prop 2.1], it can be shown, via a
+functional analytic argument, that almost sure filter stability in the total variation
+sense (which can be relaxed to weak merging) and the uniqueness of an invariant prob-
+ability measure on the state process leads to unique ergodicity of the filter process.
+It thus turns out that a complementary condition, and with an alternative argument
+(e.g. to [46, Theorem 2]), can also be established: unique ergodicity of the measure-
+17
+
+ment process and a measurability condition related to filter stability leads to unique
+ergodicity.
+Remark 5.1. For the problem at hand, for the control-free setup, it is evident that
+the decision maker can always apply arg min E[c(Xk, Uk)|Y[0,k]] at any k ∈ N. This
+policy will lower bound the expected cost under any policy, including the aforemen-
+tioned policy. The conclusion is that, asymptotically, they are equivalent. The main
+novelty here is the optimality of a stationary policy for an average cost problem.
+On the realization problem for an optimal stationary policy
+Assumption 8 will not be applicable if the initialization of the process is to be
+arbitrary. Let ¯P be the (unique) invariant probability measure on {Y(−∞,t]}. If the
+initial measure P0 is such that P0 ≪ ¯P, then a result due on the ergodic theory of
+Markov chains (see Theorem C.1(ii)) shows that for every measurable bounded g:
+lim
+T →∞
+1
+T EP0[
+T −1
+�
+k=0
+g(Y(−∞,k])] =
+�
+g(y(−∞,0]) ¯P (dy(−∞,0])
+We state the following, building on Theorem C.1(ii), and the proof method of
+Theorem 3.5.
+Theorem 5.4. Let Assumptions 6 and 5 hold. Let ut = γ(y(−∞,t], Rt) be an
+optimal stationary policy (by Theorem 5.1) with Rt being an i.i.d.
+noise process.
+Let g(y(−∞,k], xk) := E[c(Xk, γ(Y(−∞,k], Rk))] where the expectation is over Rk. Let
+¯P denote an invariant process measure on (Y(−∞,k−1], Xk) and let P0 be the invari-
+ant measure on {Y(−∞,−1]} and µ be the prior measure on X0 as imposed by the
+controller/decision maker. If P0 × µ ≪ ¯P; that is, if the (filter) initialization is inde-
+pendent of the state process initialization and under the absolute continuity condition,
+lim
+T →∞
+1
+T EP0×µ[
+T −1
+�
+k=0
+g(Y(−∞,k], Xk)] =
+�
+g(y(−∞,0], x0) ¯Q(dy(−∞,0], x0).
+(5.1)
+The optimal control/decision cost will be attained.
+Observe again, as in the controlled setup, that in the analysis above we con-
+structed y(−∞,−1] and not y(−∞,0], since µ on X0 induces directly a measure on Y0.
+Note that with P0 × µ viewed as an initial measure, when the process evolves, the
+evolution is the correct one consistent with the actual realizations under the true
+measure and thus the initialization does not affect the evolution of the measure: X0
+determines the probability of the future events: the transition kernel:
+P(dzt+1|zt) = Q(dyt+1|xt+1)P(dxt+1|xt)δy(−∞,t](dy(−∞,t])
+is such that Yt is generated according to both the true process and the actual process.
+Note that if ¯P and P0 are both stationary, we have that ¯P decomposes as
+P0(dy)Q(dx0|y(−∞,−1]). Thus, as in the controlled-case, what we need is that µ(dx0) ≪
+Q(dx0|y(−∞,−1]) P0 a.e., so that convergence in the sense of (5.1) holds.
+Appendix A. Proofs.
+A.1. Proof of Theorem 1.2. Following [2, 11], we study the limit distribution
+of the following occupation measures, under any policy γ in Γ. Let for T ≥ 1
+vT (D) = 1
+T
+T −1
+�
+t=0
+1{(πt,ut)∈D},
+D ∈ B(P(X) × U).
+18
+
+Consider any policy γ in Γ, π0 ∼ µ, and let for T ≥ 1,
+µT (D) = Eγ
+µ[vT (D)] = Eγ
+µ
+1
+T
+� T −1
+�
+t=0
+1{πt,ut)∈D}
+�
+,
+D ∈ B(P(X) × U)
+Let for µ ∈ P(X), µP(A) := � µ(dπ, du)P(πt+1 ∈ A|πt = π, ut = u). Then through
+what is often referred to as a Krylov-Bogoliubov-type argument, for every Borel A ⊂
+P(X)
+|µT (A × U) − µT P(A)| = Eγ
+µ0
+1
+T
+� T −1
+�
+t=0
+1{πt,ut)∈(A×U)} −
+T −1
+�
+t=0
+1{πt+1,ut+1)∈(A×U)}
+�
+≤ 1
+T → 0,
+as T → ∞. Notice that the above applies for any policy γ ∈ Γ. Now, if we can
+ensure that, for some subsequence µtk, µtk → µ weakly for some probability measure
+µ (which holds by the assumption that the state and action spaces are compact, which
+makes the set of probability measures on the state weakly compact), it would follow
+that µtkη(A × U) → µη(A × U). It would follow that µtkP → νP also, by the weak
+Feller condition, and hence ν = νP and ν would be stationary.
+Define, with ΓS denoting the set of stationary control policies mapping the belief
+state to actions,
+G = {v ∈ P(P(X) × U) :∃γ ∈ ΓS, v(A) =
+�
+π,u
+P γ((πt+1, ut+1) ∈ A|x)v(dx, du),
+A ∈ B(P(X) × U)}
+(A.1)
+Thus, every weakly converging sequence µt will satisfy the above equation, and
+therefore, under any admissible policy, every converging occupation measure sequence
+converges to the set G. Let us define
+γ∗ = inf
+v∈G
+�
+v(dπ, du)˜c(π, u)
+Since the expression
+�
+v(dx, du)c(x, u) is lower semi-continuous in v, a compact-
+ness condition on G will ensure the existence of an optimal occupation measure which
+is in G: we now show that there exists an optimal occupation measure in G if the tran-
+sition kernel is weak-Feller and G is weakly compact: The problem has now reduced
+to
+inf
+µ∈G
+�
+µ(dx, du)c(x, u),
+The set G is closed, since if νn → ν and νn ∈ G, then for continuous and bounded f ∈
+Cb(P(X)), ⟨νn, f⟩ → ⟨ν, f⟩. By weak-Feller continuity of the kernel
+�
+f(π′)η(dπ′|π, u)
+is also continuous and thus, ⟨νn, ηf⟩ → ⟨ν, ηf⟩ = ⟨νη, f⟩. Thus, ν(f) = νη(f) and ν ∈
+G. Therefore, G is weakly sequentially compact. Since the integral
+�
+µ(dx, du)c(x, u)
+is lower semi-continuous on the set of measures under weak convergence, and the ex-
+istence result follows from Weierstrass’ Theorem. As a result, there exists an optimal
+occupation measure, say v(dπ, du). This defines a stationary control policy by the
+Radon-Nikodym derivative: µ(u ∈ ·|π) =
+dv(dπ,·)
+d
+�
+u v(dπ,·)(π), for v a.e. π.
+19
+
+A.2. Proof of Lemma 3.1. a) Let us recall the w-s topology [3, 56]: Let A, B
+be complete, separable, metric spaces. The w-s topology on the set of probability
+measures P(A×B) is the coarsest topology under which
+�
+f(a, b)ν(da, db) : P(A×B) →
+R is continuous for every measurable and bounded f which is continuous in b ∈ B for
+every a ∈ A (but unlike weak topology, f does not need to be continuous in a).
+For every n, we have that for every function g that is continuous and bounded:
+�
+Pn(dy1, dy2, du2)g(y1, y2, u2) =
+�
+Pn(dy1|y2)Pn(du2, y2)g(y1, y2, u2)
+=
+�
+P(dy1|y2)Pn(du2, y2)g(y1, y2, u2)
+(A.2)
+Testing the equality above on continuous and bounded functions implies this
+property for any measurable and bounded function (that is, continuous and bounded
+functions form a separating class, see e.g.
+p.
+13 in [6] or Theorem 3.4.5 in [20])
+for weak convergence of probability measures. Since the marginals on y1, y2 is fixed,
+[56, Theorem 3.10] (see also [3, Theorem 2.5]) establishes that the sequence {Pn} is
+relatively compact under the w-s topology under the stated tightness condition; {Pn}
+is tight by Prohorov’s theorem.
+Now, taking the limit of both sides in (A.2), we have that the left hand side
+converges to
+�
+P(dy1, dy2, du2)g(y1, y2, u2). The right-hand side, on the other hand
+can be written as:
+� � �
+P(dy1|dy2)g(y1, y2, u2)
+�
+Pn(du2, dy2)
+(A.3)
+The expression
+� �
+P(dy1|dy2)g(y1, y2, u2)
+�
+is measurable in y2 and continuous in
+u2 by an application of the dominated convergence theorem. As a result, by the w-s
+convergence, in the limit we have
+� � �
+P(dy1|dy2)g(y1, y2, u2)
+�
+P(du2, y2)
+(A.4)
+and thus the conditional independence property is satisfied.
+b) For every n, we again have that for every function g continuous and bounded:
+�
+Pn(dy1, dy2, du2)g(y1, y2, u2) =
+�
+Pn(dy1|y2)Pn(du2, y2)g(y1, y2, u2)
+=
+�
+κ(dy1|y2)Pn(du2, y2)g(y1, y2, u2)
+(A.5)
+As in (i), testing the equality above on continuous and bounded functions implies
+this property for any measurable and bounded function [6, p. 13] or [20, Theorem
+3.4.5]) for weak convergence of probability measures. Since the marginals on y1, y2 is
+fixed, [56, Theorem 3.10] (see also [3, Theorem 2.5]) establishes that the sequence {Pn}
+is relatively compact under the w-s topology under the stated tightness condition;
+{Pn} is tight by Prohorov’s theorem. Now, taking the limit of both sides in (A.5), we
+have that the left hand side converges to
+�
+P(dy1, dy2, du2)g(y1, y2, u2). The right-
+hand side can be written as:
+� � �
+κ(dy1|dy2)g(y1, y2, u2)
+�
+Pn(du2, y2)
+(A.6)
+20
+
+The expression
+� � κ(dy1|dy2)g(y1, y2, u2)
+�
+is measurable in y2 and continuous in u2
+by an application of dominated convergence theorem. Furthermore, by the condition
+that κ is a continuous kernel, we have that
+� �
+κ(dy1|dy2)g(y1, y2, u2)
+�
+is continuous
+in y2, by an application of a generalized dominated convergence theorem [57, Theorem
+3.5] or [44, Theorem 3.5]. As a result, in the limit we have
+� � �
+P(dy1|dy2)g(y1, y2, u2)
+�
+P(du2, y2)
+(A.7)
+and thus the conditional independence property is satisfied.
+A.3. Proof of Theorem 3.3. Recall (A.1) and let
+µT (D) = E[vT (D)] = Ev0
+1
+T
+�
+T
+�
+t=1
+1{zt,ut)∈D}
+�
+,
+D ∈ B(X × YZ− × UZ− × U).
+This can also be written as 1
+T
+� �T
+t=1 P({(zt, ut) ∈ D})
+�
+. Observe that for any t,
+xt ↔ y(−∞,t] ↔ ut holds. Let (xt, y(−∞,t], ut) ∼ Pt. As in the proof of Theorem A.1,
+through a Krylov-Bogoliubov-type argument, for every Borel A ∈ B(X × YZ− × UZ−
+|µN(A × U) − µNP(A)|
+=
+����
+1
+N
+�
+(µ0(A × U) + · · · + µP(N−1)(A)) − (mu0P(A) + · · · + vPN(A))
+�����
+≤ 1
+N |µ0(A × U) − µ0PN(A)| → 0.
+(A.8)
+As earlier, in the proof of Theorem A.1, if we can ensure that for some subsequence,
+µtk → µ for some probability measure µ, it would follow that µtkP → µ also and
+hence µ = µP and µ would be invariant and µ ∈ G, where
+G = {v ∈ P(X × YZ− × UZ−) :
+v(B × U) =
+�
+x,u
+P(zt+1 ∈ B|z, u)v(dz, du), B ∈ B(X × YZ− × UZ−)}
+Convergence to G is by (A.8) is a consequence of Lemma 2.3 under Assumption 4.
+We will show that, if convergence occurs, it must also be that µ ∈ H;
+H = {v ∈ P(X × YZ− × UZ−) :
+v(dx0, dy(−∞,0], du(−∞,−1], du0)
+= v(dx0, dy(−∞,0], du(−∞,−1])v(du0|dy(−∞,0], du(−∞,−1])},
+(A.9)
+that is the control action variables are conditionally independent from the state vari-
+ables given the information variable y(−∞,0], u(−∞,−1], thus satisfying (2.5).
+We show now that µt, possibly along a subsequence, will also converge to H.
+Observe that
+�
+µT (dx, dy(−∞,0], du(−∞,−1], du)f(x, y(−∞,0], u(−∞,−1], u)
+21
+
+=
+�
+1
+T
+T
+�
+t=1
+Pt(dx, dy(−∞,0], du(−∞,−1], du0)f(x, y(−∞,0], u(−∞,−1], u0)
+=
+�
+1
+T
+T
+�
+t=1
+Pt(dx|y(−∞,0], du(−∞,−1])Pt(dy(−∞,0], du(−∞,−1], du0)f(x, y(−∞,0], u(−∞,−1], u0)
+=
+�
+1
+T
+T
+�
+t=1
+�
+Pt(dx|y(−∞,0], du(−∞,−1])f(x, y(−∞,0], u(−∞,−1], u0)
+�
+Pt(dy(−∞,0], du(−∞,−1], du0)
+=
+�
+1
+T
+T
+�
+t=1
+�
+P(dx|y(−∞,0], du(−∞,−1])f(x, y(−∞,0], u(−∞,−1], u0)
+�
+Pt(dy(−∞,0], du(−∞,−1], du0)
+(A.10)
+=
+�
+1
+T
+T
+�
+t=1
+�
+g(y(−∞,0], u(−∞,−1], u0)
+�
+Pt(dy(−∞,t], du(−∞,t−1], dut)
+(A.11)
+=
+�
+1
+T
+T
+�
+t=1
+�
+g(y(−∞,t], u(−∞,t−1], ut)
+�
+Pt(dy(−∞,t], du(−∞,t−1], dut)
+→
+� �
+g(y(−∞,0], u(−∞,−1], u0)
+�
+µ(dy(−∞,0], du(−∞,−1], du0)
+(A.12)
+=
+� �
+P(dx|y(−∞,0], u(−∞,−1])f(x, y(−∞,0], u(−∞,−1], u0)
+�
+µ(dy(−∞,0], du(−∞,−1], du0)
+=
+� �
+P(dx|y(−∞,0], u(−∞,−1])µ(dy(−∞,0], du(−∞,−1], du0)
+�
+f(x, y(−∞,0], u(−∞,−1], u0)
+(A.13)
+Here (A.10) is due to Assumption 61 and in (A.11) we define
+g(y(−∞,0], u(−∞,−1], u0) =
+�
+P(dx|y(−∞,0], u(−∞,−1])f(x, y(−∞,0], u(−∞,−1], u0)
+�
+.
+Here, (A.12) follows from weak continuity of the kernel under Assumption 5. Thus, if
+µt(·) converges weakly to µ(·), µ satisfies the conditional independence property and
+is thus in H.
+In view of this, the existence problem reduces to
+inf
+v∈G∩H
+�
+v(dx, dy(−∞,0], du(−∞,−1], du)c(x, u)
+(A.14)
+By Lemma 3.2 we have that G ∩ H is a compact set under weak topology, and
+since c is continuous, there exists an optimal measure v ∈ G ∩ H.
+A.4. Proof of Theorem 3.5. Let ¯P instead denote an invariant process mea-
+sure on (Y(−∞,k], U(−∞,k−1], Uk, Xk). Let P0 be the projection of ¯P on dy(−∞,−1], du(−∞,−1].
+If P0(dy(−∞,−1], du(−∞,−1])×π0(dx0, dy0, duu) ≪ ¯P; that is, with the initial state dis-
+tribution selected independently of the past process; then with ut = γ(y(−∞,t], u(−∞,t−1], rt)
+1In the absence of Assumption 6 (A.10) would be incorrect; as a counterexample, consider yk
+giving no information (e.g. yk being constant); in this case, we clearly require Xt to be stationary
+for this to be correct since there is no information: P (Xt|Y−∞,t]) = P (Xt+1|Y−∞,t+1]).
+22
+
+and
+g(y(−∞,k], u(−∞,k−1], xk) = E[c(xk, uk)|xk, y(−∞,k], u(−∞,k−1]],
+lim
+T →∞
+1
+T EP0×π0
+T −1
+�
+k=0
+g(y(−∞,k], u(−∞,k−1], xk) =
+�
+g(y(−∞,k], u(−∞,k−1], xk) ¯Q(y(−∞,0], u(−∞,−1])
+for some ¯Q which is invariant. Note here that ¯Q does not need to be equal to ¯P.
+A.5. Proof of Theorem 5.1. The proof closely follows that of Theorem 3.3.
+Recall (A.1) and let
+µT (D) = E[vT (D)] = Ev0
+1
+T
+�
+T
+�
+t=1
+1{zt,ut)∈D}
+�
+,
+D ∈ B(X × YZ− × UZ− × U).
+This can also be written as
+1
+T
+�
+T
+�
+t=1
+P({(zt, ut) ∈ D})
+�
+Observe that for any t, xt ↔ y(−∞,t] ↔ ut holds. Let (xt, y(−∞,t], ut) ∼ Pt. Now,
+again through the Krylov-Bogoliubov-type argument, for every Borel A
+|µN(A × U) − µNP(A)|
+=
+����
+1
+N
+�
+(µ0(A × U) + · · · + µP(N−1)(A)) − (v0P(A) + · · · + vPN(A))
+�����
+≤ 1
+N |µ0(A × U) − µ0PN(A)| → 0.
+(A.15)
+Now, if we can ensure that for some subsequence, µtk → µ for some probability
+measure µ, it would follow that µtkP(B) → µ(B) also and hence µ(B) = µP(B) and
+µ would be invariant.
+Thus, if µt converges, it must be that µt → µ for some µ ∈ G ∩ H, where
+G = {v ∈ P(X × YZ− × U) :
+v(B × U) =
+�
+x,u
+P(X1, Y(−∞,1] ∈ B|x0, y(−∞,0], u)v(dx0, dy(−∞,0], du),
+B ∈ B(X × YZ− × U)}
+(A.16)
+and
+H = {v ∈ P(X × YZ− × U) :
+v(dx, dy(−∞,0], du0) = v(dx, dy(−∞,0])P(du0|dy(−∞,0])},
+(A.17)
+that is the control action variables are conditionally independent from the state vari-
+ables given the information variable y(−∞,0], u(−∞,−1].
+Convergence to G is by (A.15) as a consequence of Lemma 2.3 under Assumption
+4.
+23
+
+We show now that µt will also converge to H.
+Case 1. Under Assumption 8. For a continuous and bounded f, if µt → µ
+for some µ, we have that
+�
+µt(dx, dy(−∞,0], du)f(x, y(−∞,0], u)
+=
+�
+1
+T
+T
+�
+t=1
+Pt(dx, dy(−∞,0], du0)f(x, y(−∞,0], u0)
+=
+�
+1
+T
+T
+�
+t=1
+Pt(dx|y(−∞,0])Pt(dy(−∞,0], du0)f(x, y(−∞,0], u0)
+=
+�
+1
+T
+T
+�
+t=1
+�
+Pt(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+Pt(dy(−∞,0], du0)
+=
+�
+1
+T
+T
+�
+t=1
+�
+P(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+Pt(dy(−∞,0], du0)
+(A.18)
+=
+�
+1
+T
+T
+�
+t=1
+�
+g(y(−∞,0], u0)
+�
+Pt(dy(−∞,t], dut)
+(A.19)
+=
+�
+1
+T
+T
+�
+t=1
+�
+g(y(−∞,t], ut)
+�
+Pt(dy(−∞,t], dut)
+→
+� �
+g(y(−∞,0], u0))
+�
+µ(dy(−∞,0], du0)
+=
+� �
+P(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+µ(dy(−∞,0], du0)
+=
+� �
+P(dx|y(−∞,0])µ(dy(−∞,0], du0)
+�
+f(x, y(−∞,0], u0)
+(A.20)
+where (A.18) uses stationarity and in (A.19) we define
+g(y(−∞,0], u0) =
+�
+P(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+.
+This expression will converge as soon as µt(dy(−∞,0], du0) converges weakly to µ,
+where µ satisfies the conditional independence property. Here, µt converges to µ in
+the w −s sense. Thus, even in the absence of Assumption 5, convergence holds in this
+case. 2
+Case 2. Under Assumptions 5 and 6. If we don’t assume stationarity, we can
+modify the above as follows, but then we need the weak continuity condition stated
+in Assumption 5:
+�
+µt(dx, dy(−∞,0], du)f(x, y(−∞,0], u)
+2Here, we crucially assume that the marginal on xk, y(−∞,k] is fixed, and the realizations are
+generated according to the stationary measure. This is crucial for the argument in (A.18). In the
+above, we don’t need weak continuity.
+24
+
+=
+�
+1
+T
+T
+�
+t=1
+Pt(dx, dy(−∞,0], du0)f(x, y(−∞,0], u0)
+=
+�
+1
+T
+T
+�
+t=1
+Pt(dx|y(−∞,0])Pt(dy(−∞,0], du0)f(x, y(−∞,0], u0)
+=
+�
+1
+T
+T
+�
+t=1
+�
+Pt(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+Pt(dy(−∞,0], du0)
+=
+�
+1
+T
+T
+�
+t=1
+�
+P(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+Pt(dy(−∞,0], du0)
+(A.21)
+=
+�
+1
+T
+T
+�
+t=1
+�
+g(y(−∞,0], u0)
+�
+Pt(dy(−∞,t], dut)
+(A.22)
+=
+�
+1
+T
+T
+�
+t=1
+�
+g(y(−∞,t], ut)
+�
+Pt(dy(−∞,t], dut)
+→
+� �
+g(y(−∞,0], u0))
+�
+µ(dy(−∞,0], du0)
+(A.23)
+=
+� �
+P(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+µ(dy(−∞,0], du0)
+=
+� �
+P(dx|y(−∞,0])µ(dy(−∞,0], du0)
+�
+f(x, y(−∞,0], u0)
+(A.24)
+3 Here (A.21) is due to Assumption 6 and in (A.22) we define
+g(y(−∞,0], u0) =
+�
+P(dx|y(−∞,0])f(x, y(−∞,0], u0)
+�
+.
+This expression will converge as soon as µt(dy(−∞,0], du0) converges weakly to µ,
+where µ satisfies the conditional independence property. Here, (A.23) follows from
+weak continuity of the kernel under Assumption 5.
+In view of this, the existence problem reduces to
+inf
+v∈G∩H
+�
+v(dx, dy(−∞,0], du)c(x, u)
+where
+We note that under weak continuity of the transition kernel for P, G is a closed
+set under weak convergence. By Lemma 3.1, H is also closed. Since the state space
+is compact, the set of probability measures on (X × YZ− × U) is tight. Thus G ∩ H is
+a compact set under weak topology and since c is continuous, there exists an optimal
+measure v ∈ G ∩ H.
+Appendix B. Discussion on Assumptions 5, 7 and 6 .
+3In the absence of Assumption 6 (A.21) would be incorrect. As a counterexample, consider yk =
+no information; in this case, we clearly require Xt to be stationary for this to be correct since there
+is no information: P (Xt|Y−∞,t]) = P (Xt+1|Y−∞,t+1]).
+25
+
+B.1. Sufficient Conditions for Assumption 5. For the case with finite mea-
+surement and actions, this condition is satisfied by almost sure filter stability, though
+with a uniformity condition over priors, in the following sense
+sup
+ν ∥πµ
+n − πν
+n∥BL → 0 P µa.s.,
+(B.1)
+where BL denotes the bounded Lipschitz norm (or any other weak convergence in-
+ducing metric can be used).
+This condition (including uniformity) holds under a
+contraction analysis via the Hilbert metric, as shown in [27, Corollary 4.2]. Further-
+more, by modifying the proof of [47, Lemma 3.5] to place supν inside the expectation,
+[47, Corollary 3.7] also leads to this condition in view of Borel-Cantelli Lemma as
+noted in [47, Remark 3.10].
+The sufficiency of (B.1) builds on the following: Observe that
+E[f(X0)|yn
+(−∞,0], un
+(−∞,−1]] − E[f(X0)|y(−∞,0], u(−∞,−1]]
+= E[f(X0)|yn
+(−∞,0], un
+(−∞,−1]] − E[f(X0)|(y, u)n
+(−∞,−N], y(−N+1,0], u(−N+1,−1]]
+(B.2)
++E[f(X0)|(y, u)n
+(−∞,−N], y(−N+1,0], u(−N+1,−1]] − E[f(X0)|y(−∞,0], u(−∞,−1]]
+(B.3)
+The uniform filter stability condition (B.1) will allow us to truncate the past finite
+window: For every ǫ > 0 and π = P(Xn ∈ ·|y(∞,n−, u(∞,n−1) select N so that, the
+effect of the history is uniformly, over all priors P(X−N ∈ ·|(y, u)n
+(−∞,−N]), is less
+than ǫ
+2, so that (B.3) is bounded uniformly over all sequences prior to −N.
+We then apply a continuity argument for the first term, (B.2):
+For this first term, in case the the measurements and actions are finitely valued,
+the desired result follows since for sufficiently large n, the first N coordinates of mea-
+surement and actions y(−N+1,0], u(−N+1,−1], will need to match to satisfy proximity
+under the product metric so that for all sufficiently large n:
+yn
+(−N+1,0], un
+(−N+1,−1] = y(−N+1,0], u(−N+1,−1],
+making (B.2) zero.
+For the case with continuous measurements and actions, conditions in [48, Lemma
+4.6] suffices, together with the uniform filter stability condition presented above above.
+B.2. Sufficient Conditions for Assumptions 6 and 7. In the control-free
+case; Assumption 6 is implied by stationarity or both Assumptions 7 and 6 hold un-
+der almost sure filter stability, see e.g. [17, 16]. On Assumption 7, as noted earlier a
+further related sufficient condition, obtained via the Hilbert metric, is [27, Corollary
+4.2]. Complementing this, for both controlled and control-free setups, conditions in
+[47] (via [42, Theorem 2, Part 2] leading also to almost sure stability under total varia-
+tion) based on Dobrushin’s coefficients of the measurement channel and the controlled
+transition kernel leads to almost sure filter stability and accordingly Assumption 6.
+Appendix C. An ergodic theorems for Markov chains.
+Suppose that {Xt}t≥0 denote a discrete-time Markov chain with state space X, a
+Polish space.
+Theorem C.1. [31] [63] Let ¯P be an invariant probability measure for a Markov
+process.
+26
+
+(i) [Ergodic decomposition and weak convergence] For x, ¯P a.s., 1
+N Ex[�N−1
+t=0 1{xn∈·}] →
+Px(·) weakly and ¯P is invariant for Px(·) in the sense that
+¯P(B) =
+�
+Px(B) ¯P (dx)
+(ii) [Convergence in total variation] For all µ ∈ P(XN) which satisfies that µ ≪ ¯P
+(that is, µ is absolutely continuous with respect to ¯P), there exists v∗ such
+that
+∥Eµ[ 1
+N
+N−1
+�
+t=0
+1{T nX∈·}] − v∗(·)∥T V → 0.
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+MNRAS 000, 1–22 (2022)
+Preprint 6 January 2023
+Compiled using MNRAS LATEX style file v3.0
+The Co-Ordinated Radio and Infrared Survey for High-Mass Star
+Formation. V. The CORNISH-South Survey and Catalogue.
+T. Irabor,1★ M.G. Hoare,1 M. Burton,13 W.D. Cotton,3 P. Diamond,2 S. Dougherty,21
+S.P. Ellingsen,15 R. Fender,14 G.A. Fuller,2,20 S. Garrington, 2 P.F. Goldsmith,5 J. Green,12 A.G. Gunn2
+J. Jackson,7 S. Kurtz,4 S.L. Lumsden,1 J. Marti,11 I. McDonald,2,22 S. Molinari,16 T.J. Moore,8
+M. Mutale,1 T. Muxlow, 2 T. O’Brien, 2 R.D. Oudmaijer,1 R. Paladini,19 J.D. Pandian,6 J.M. Paredes,10
+A.M.S. Richards, 2 A. Sanchez-Monge, 20 R. Spencer, 2 M.A. Thompson,1,9 G. Umana,18 J.S. Urquhart,17
+M. Wieringa,12 and A. Zijlstra, 2
+1Physics and Astronomy, University of Leeds, LS2 9JT, UK
+2Jodrell Bank Centre for Astrophysics, Department of Physics and Astronomy, University of Manchester, Manchester, M13 9PL, UK
+3The National Radio Astronomy Observatory, Charlottesville, VA 22903, USA
+4Institute of Radio Astronomy and Astrophysics, National Autonomous University of Mexico, 58089 Morelia, Michoacán, México
+5Jet Propulsion Laboratory California Institute fo Technology, Pasadena CA, 91109
+6Department of Earth & Space Sciences, Indian Institute of Space Science and Technology, Trivandrum 695547, India
+7Green Bank Observatory, 155 Observatory Rd, P.O. Box 2, Green Bank, WV 24944, USA
+8Astrophysics Research Institute, Liverpool John Moores University, Twelve Quays House, Egerton Wharf, CH41 1LD, UK
+9Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield AL10 9AB, UK
+10Cosmos Science Institute, University of Barcelona , ICCUB, Martí i Franqués, 1, 08028 Barcelona, Spain
+11Departamento de Física (EPSJ), Universidad de Jaén, Campus Las Lagunillas s/n, A3, E-23071 Jaén, Spain
+12CSIRO Space and Astronomy, PO Box 1130, Bentley WA 6102, Australia
+13Armagh Observatory and Planetarium,College Hill, BT61 9DB, Northern Ireland
+14Department of Physics, University of Oxford, Keble Road, Oxford, OX1 3RH, UK
+15School of natural Sciences, College of Sciences and Engineering, University of Tasmania, Hobart 7001, TAS, Australia
+16Istituto Nazionale di Astrofisica - IAPS, Via Fosso del Cavaliere 100, I-00133 Roma, Italy
+17Centre for Astrophysics and Planetary Science, University of Kent, Canterbury, CT2 7NH, UK
+18INAF - Osservatorio Astrofisico di Catania, Via S. Sofia 78, I-95123, Catania, Italy
+19Infrared Processing Center, California Institute of Technology, Pasadena, CA 91125, USA
+20Physikalisches Institut, University of Cologne, Zülpicher Str. 77, 50937 Köln, Germany
+21The ALMA headquarters, Santiago, Alonso de Córdova 3107, Chile
+22Department of Physics and Astronomy, Open University, Walton Hall, Milton Keynes, MK7 6AA, UK
+Accepted XXX. Received YYY; in original form ZZZ
+ABSTRACT
+We present the first high spatial resolution radio continuum survey of the southern Galactic plane. The CORNISH project
+has mapped the region defined by 295◦ < l < 350◦; |b| < 1◦ at 5.5-GHz, with a resolution of 2.5′′ (FWHM). As with the
+CORNISH-North survey, this is designed to primarily provide matching radio data to the Spitzer GLIMPSE survey region.
+The CORNISH-South survey achieved a root mean square noise level of ∼ 0.11 mJy beam−1, using the 6A configuration of
+the Australia Telescope Compact Array (ATCA). In this paper, we discuss the observations, data processing and measurements
+of the source properties. Above a 7𝜎 detection limit, 4701 sources were detected, and their ensemble properties show similar
+distributions with their northern counterparts. The catalogue is highly reliable and is complete to 90 per cent at a flux density
+level of 1.1 mJy. We developed a new way of measuring the integrated flux densities and angular sizes of non-Gaussian sources.
+The catalogue primarily provides positions, flux density measurements and angular sizes. All sources with IR counterparts at
+8𝜇m have been visually classified, utilizing additional imaging data from optical, near-IR, mid-IR, far-IR and sub-millimetre
+galactic plane surveys. This has resulted in the detection of 524 H II regions of which 255 are ultra-compact H II regions, 287
+planetary nebulae, 79 radio stars and 6 massive young stellar objects. The rest of the sources are likely to be extra-galactic. These
+data are particularly important in the characterization and population studies of compact ionized sources such as UCHII regions
+and PNe towards the Galactic mid-plane.
+Key words: catalogues < Astronomical Data bases – (ISM:) H II regions < Interstellar Medium (ISM), Nebulae – radio
+continuum: ISM < Resolved and unresolved sources as a function of wavelength – surveys < Astronomical Data bases –
+techniques: image processing < Astronomical instrumentation, methods, and techniques
+★ E-mail: [email protected] (TI)
+© 2022 The Authors
+arXiv:2301.01988v1  [astro-ph.GA]  5 Jan 2023
+
+2
+T. Irabor et al.
+1 INTRODUCTION
+Understanding the formation and evolution of the content of our
+Galaxy requires studying an unbiased population of objects covering
+different evolutionary stages utilizing a wide range of wavebands.
+Cm-wave radio continuum surveys are useful for probing the ionized
+gas components such as H II regions and planetary nebulae.
+The ultra-compact (UC) HII population provides a means to probe
+the early phases of massive star formation, where the young stars
+are still deeply embedded in their natal molecular clouds. They are
+characterized by physical sizes ≤ 0.1 pc, high emission measures
+≥ 107 pc cm−6, high electron densities ≥ 104 cm−3 and lifetimes
+typical of 105 yrs (Wood & Churchwell 1989; Comeron & Torra
+1996; Kurtz 2000; Kurtz & Franco 2002; Churchwell 2002). Given
+that these populations often form in clusters, radio and infrared
+observations at arcsecond resolution are required to resolve their
+morphologies and provide insight into their immediate environment
+(Churchwell 1990; Hoare et al. 2007). An unbiased survey of the pop-
+ulation of UCHII regions will allow us to test evolutionary models
+of massive star formation, H II region dynamics, galactic structure
+and massive star formation rate in our Galaxy (Churchwell 2002;
+Hoare et al. 2007; Davies et al. 2011; Urquhart et al. 2013b; Steggles
+2016). Radio observations of UCHII regions are also more useful
+when carried out at a frequency where thermal free-free emission
+is optically thin (≥5-GHz: Churchwell 1990; Wood & Churchwell
+1989).
+IR surveys with high sensitivity and arcsecond resolution of the
+Galactic plane have made possible studies of an unbiased and statisti-
+cally representative population of Galactic objects, thereby aiding the
+studies of massive star formation and stellar evolution. In the northern
+Galactic plane, these surveys include the the mid-infrared Galactic
+Legacy Infrared Mid-Plane Survey Extraordinaire (GLIMPSE) by
+the Spitzer satellite (Churchwell et al. 2009), the mid-infrared inner
+Galactic plane survey using the Multiband Infrared Photometer for
+Spitzer (MIPSGAL1: Carey et al. 2009), the far-infrared Herschel
+Infrared Galactic Plane (Hi-GAL) survey (Molinari et al. 2010), the
+near-infrared Galactic plane survey (GPS) of the United Kingdom
+Infrared Deep Sky Survey (UKIDSS) project (Lucas et al. 2008),
+sub-millimetre APEX Telescope Large Area Survey of the Galaxy
+(ATLASGAL: Schuller et al. 2009) and the H𝛼 Isaac Newton Tele-
+scope Photometric Survey (IPHAS: Drew et al. 2005).
+To complement these surveys, the CORNISH project delivered a
+uniform and high resolution (1.5′′) radio continuum dataset of the
+northern Galactic plane at 5-GHz. It achieved a sensitivity ∼ 0.43
+mJy beam−1 using the VLA in B and BnA configurations (Hoare
+et al. 2012; Purcell et al. 2013: hereafter Paper I and Paper II, re-
+spectively). These data made possible an unbiased census of UCHII
+regions that is the largest selection to date in the northern Galactic
+plane (Kalcheva et al. 2018: hereafter Paper III). Kalcheva (2018)
+found that 70 per cent of UCHII regions had a cometary morphology
+using the CORNISH data and follow-up higher resolution radio data.
+Through a multi-wavelength analysis, Irabor et al. (2018) (hereafter
+Paper IV) uncovered an unbiased population of compact planetary
+nebulae (PNe), of which 7 per cent were newly discovered PNe. A
+subset of the PNe population has properties that are typical of young
+sources (Paper IV and Fragkou et al. 2018). The study of such objects
+is critical, given that the transition window from the post-AGB to the
+PN phase is small (usually < 1000 yrs). Other studies that used the
+CORNISH data in massive star formation studies include Urquhart
+et al. (2013a); Cesaroni et al. (2015); Tremblay et al. (2015); Yang
+1 http://mipsgal.ipac.caltech.edu/
+et al. (2019) and Djordjevic et al. (2019). Note that the Global view
+on Star formation in the Milky Way (GLOSTAR) Galactic plane sur-
+vey (Brunthaler et al. 2021), is using the JVLA to go deeper than the
+CORNISH survey of the northern Galactic plane.
+In the southern Galactic plane, the VST Photometric H𝛼 Survey
+(VPHAS+: Drew et al. 2014) and the Vista Variables in the Via Lactea
+(VVV) survey (Minniti et al. 2011) delivered high resolution H𝛼 and
+near-infrared data respectively, in addition to existing infrared and
+sub-millimetre data (see Table 1). Surveys of masers at radio wave-
+lengths such as the Methanol Multibeam Survey (MMB; Green et al.
+2012, 2017) and the H2O Southern Galactic Plane Survey (HOPS;
+Walsh et al. 2011) are useful tracers of massive star formation. Exist-
+ing radio continuum surveys include the Molongolo Galactic Plane
+Survey (MGPS) (Murphy et al. 2007; Green 1999) that surveyed the
+245◦ < l < 365◦ and |b| ≤ 10◦ region at 843 MHz with a resolution
+of 45′′, the 1.4-GHz Southern Galactic Plane Survey (SGPS) that
+mapped the 253◦ < l < 358◦; |b| < 1◦.5 region, with a resolution
+of 100′′ and sensitivity of 1 mJy beam−1 (McClure-Griffiths et al.
+2005; Haverkorn et al. 2006), the GaLactic and Extragalactic All-
+sky Murchison Widefield Array (GLEAM) survey at 72–231 MHz,
+with a resolution of 4′–2′ that covers all the southern plane (Hurley-
+Walker et al. 2019) and the TIFR GMRT Sky Survey (TGSS) survey
+at 150 MHz and 25′′ resolution covering the sky above declination
+−53◦ (Intema et al. 2017). These surveys are too low a resolution to
+resolve UCHII regions and are at frequencies where these objects are
+optically thick.
+In this paper we present a new radio continuum survey from the
+CORNISH project of the southern Galactic Plane. The observations
+are described in Section 2. The calibration and imaging are discussed
+in Sections 3 and 4. Data quality and measurement of source prop-
+erties are presented in Sections 5 and 6 along with a discussion of
+the catalogue and its reliability. Furthermore, example sources and
+a comparison of the CORNISH catalogue with other surveys are
+presented in Section 6.
+2 OBSERVATIONS
+The CORNISH program observed the southern Galactic plane with
+the Australia Telescope Compact Array (ATCA2) using the 2-GHz
+bandwidth of the Compact Array Broadband Backend (CABB) cor-
+relator (Wilson et al. 2011). Observations were carried out with the
+6A3 array configuration for about 400 hours at two different fre-
+quency bands (4.5 - 6.5-GHz and 8 - 10-GHz) in full-polarization,
+centred on 5.5-GHz and 9-GHz, simultaneously. This paper focuses
+on the 4.5 - 6.5-GHz band. Data reduction of the 8 - 10-GHz data is
+still on-going and will be presented in a future paper (T. Irabor et al.,
+In preparation). Observations were between 2010 and 2012 and the
+observation parameters are summarized in Table 2.
+The survey utilized on-the-fly mapping, such that the antennas
+were scanning continuously, while the phase centres were sequen-
+tially moved in a traditional mosaic pattern. This resulted in the
+doubling of the uv-coverage in a single run and a primary beam
+that is elongated in the scanning direction4. The phase centres were
+spaced at 7.4′ in a mosaic pattern that is a scaled version of the
+hexagonal mosaic implemented in the NVSS survey (Condon et al.
+2 https:/www.narrabri.atnf.csiro.au/
+3 https://www.narrabri.atnf.csiro.au/cgi-bin/obstools/
+baselines.cgi?array=6a
+4 http://www.narrabri.atnf.csiro.au/observing/users_guide/
+html/atug.html
+MNRAS 000, 1–22 (2022)
+
+Data processing and catalogue.
+3
+Table 1. Multi-wavelength high resolution and sensitivity surveys in the southern Galactic plane.
+Survey
+Bands
+Resolution and Sensitivity
+Coverage
+Reference
+GLIMPSE
+3.6𝜇m, 4.5𝜇m, 5.8𝜇m, and 8.0𝜇m
+<2′′; 0.4 mJy at 8𝜇m (5𝜎)
+295◦ < l < 360◦; |b| ≤ 1◦
+Churchwell et al. (2009)
+MIPSGAL
+24𝜇m
+∼ 10′′; 1.3 mJy (5𝜎)
+295◦ < l < 350◦; |b| ≤ 1◦
+Carey et al. (2009)
+VVV
+J, H, Ks
+∼ 1′′; 16.5 mag in K (5𝜎)
+295◦ < l < 350◦; |b| ≤ 2◦
+Minniti et al. (2011)
+VPHAS+
+H𝛼, r, i
+∼1′′; 20.56 mag in H𝛼 (5𝜎)
++210◦ ≲ l ≲ +40◦; |b| ≤ 5◦
+Drew et al. (2014)
+Hi-Gal
+70, 160, 250, 350 and 500𝜇m
+6 − 40′′; ∼ 13-27 mJy (1𝜎)
+−70◦ ≲ l ≲ 68◦; |b| ≤ 1◦
+Molinari et al. (2010)
+ATLASGAL
+870𝜇m
+18′′, ∼50 - 70 mJy beam−1 (1𝜎)
+280◦ < l < 60◦; |b| < 1.5◦
+Schuller et al. (2009)
+Table 2. Summary of observation parameters for the CORNISH-South survey.
+Parameters
+Value
+Observation region
+295◦ < l < 350◦; |𝑏| ≤ 1◦
+Total Time
+∼ 400 hours
+Number of antennas
+6
+Number of baselines
+15
+Observation period
+2010 to 2012
+Observation frequency
+5.5-GHz
+Bandwidth
+2-GHz
+Longest baseline
+6 km
+Size of single dish
+22 m
+Field of view/ Primary beam (FWHM)
+∼ 10′
+Synthesized beam (FWHM)
+2.5′′
+Root mean square (rms) noise level
+∼ 0.11 mJy beam−1
+1998). This spacing delivers a sensitivity that is uniform to 10 per
+cent at 5.5-GHz.
+The survey region was divided into 33 blocks (33 days of ob-
+servations) covering 110 deg2 of the Galactic plane, defined by
+295◦ < l < 350◦ and |b| ≤ 1◦. The mosaic was scanned in Galac-
+tic latitude and required ∼18 pointings to cover the 2◦ of a row (i.e
+−1◦ to +1◦). At a scan rate of 7.′4 / 10 s, each row was completed in
+3.2 minutes, including the turnaround time. A secondary calibrator
+was observed for 2 mins after observations of eight rows. A further
+eight rows were then observed with a secondary calibrator to com-
+plete one uv-cut of a block of observation, summing up to 54 mins.
+To achieve an optimum uv-coverage, this was repeated eleven times
+at different LSTs, resulting in 1.8 minutes of on-source time. Twelve
+hours were spent on each block, including flux calibration and set
+up time. Sixteen rows corresponds to 1.7◦ in longitude, hence 33 of
+these 1.7◦ × 2◦ blocks were required to cover the survey area. PKS
+B1934-638 was observed at the end of each block of observations as
+the primary flux calibrator. PKS B0823-500 was also observed at the
+beginning of each block of observations as a backup flux calibrator.
+The secondary calibrators and corresponding days of observation are
+presented in Table 3.
+The observations fall naturally into two epochs (see Figure 1),
+based on the observation period (between 2010 and 2012). Fields
+that were missed or days with observation difficulties due to bad
+weather or correlator problems were repeated to achieve as full a
+coverage as possible within the allocated time.
+3 DATA REDUCTION PIPELINE
+For us to achieve a similar uniform processing as in the CORNISH-
+North survey, we implemented a similar semi-automated calibration
+and imaging pipeline (see Figure 1 in Paper II). The pipeline was
+implemented in python language using mirpy5 to directly interface
+5 https://pypi.org/project/mirpy/
+160
+180
+200
+220
+240
+260
+280
+ (deg)
+-30
+-40
+-50
+-60
+-70
+ (deg)
+1148-671
+1352-63
+1511-55
+1646-50
+1714-397
+1729-37
+Secondary Calibrator
+Epoch I
+Epoch II
+300
+310
+320
+330
+340
+350
+l (deg)
+1
+0
+1
+b (deg)
+Figure 1. Coverage of the CORNISH-South data and positions of the six
+secondary calibrators used for calibration. Epoch 1 (red shaded regions)
+is defined by block 2010-12-21 to 2011-01-07 and epoch II (black shaded
+regions) is defined by 2011-12-20 to 2012-01-07.
+with the Multichannel Image Reconstruction, Image Analysis and
+Display (MIRIAD) software packages (Sault et al. 1995).
+3.1 Flagging and Calibration
+The raw data were converted to a MIRIAD uv-file format using the
+atlod task. Known bad channels and radio-frequency interference
+(RFI) at the time of observations were flagged. The system variables
+such as uv-coverage were inspected for each 12-hour block of obser-
+vations. The system variables allowed quick identification of blocks
+with poor uv-coverage or times with bad visibilities. Additionally,
+the amplitude and phase variations with time were inspected visu-
+ally before any flagging. This was to ensure that no more data were
+flagged than necessary, to minimize any impact on image fidelity.
+The epochs (I and II) presented different flagging demands. All
+XX polarization data of all baselines to antenna ca016 were flagged
+for epoch I. For epoch II, the YY polarization data of all baselines
+to antenna ca01 were flagged instead. This was due to a ripple in the
+bandpass at the time of observation, which would lead to false struc-
+tures in the final images7. We performed both automated (pgflag) and
+manual flagging (uvflag) on the flux calibrator, secondary calibrators
+and science data.
+6 https://www.narrabri.atnf.csiro.au/observing/users_
+guide/html/chunked/aph.html
+7 https://atcaforum.atnf.csiro.au/viewtopic.php?f=11&t=184
+MNRAS 000, 1–22 (2022)
+
+4
+T. Irabor et al.
+Table 3. Secondary calibrators for each block of observations and the corresponding longitude range.
+Date
+Calibrator (s)
+Longitude Range
+Date
+Calibrator (s)
+Longitude Range
+2010−12−21
+1714−397
+348.5 - 350.0
+2011−12−21
+1511−55
+317.8 - 319.4
+2010−12−22
+1729−37
+346.8 - 348.5
+2011−12−22
+1352−63
+304.1 - 305.7
+2010−12−23
+1729−37
+345.1 - 346.8
+2011−12−23
+1511−55
+316.0 - 317.6
+2010−12−24
+1646−50
+343.4 - 345.1
+2011−12−24
+1352−63, 1511−55
+314.3 - 315.9
+2010−12−25
+1646−50
+341.7 - 343.4
+2011−12−25
+1352−63
+310.9 - 312.5
+2010−12−26
+1646−50
+340.0 - 341.7
+2011−12−26
+1352−63
+309.2 - 310.8
+2010−12−27
+1646−50
+338.3 - 340.0
+2011−12−27
+1148−671
+297.2 - 298.8
+2010−12−28
+1646−50
+336.5 - 338.3
+2011−12−28
+1148−671, 1511−55
+295.5 - 297.1
+2010−12−29
+1646−50
+334.8 - 336.5
+2011−12−29
+1148−671
+299.0 - 304.0
+2010−12−30
+1646−50
+331.4 - 333.0
+2011−12−30
+1352−63
+305.8 - 307.5
+2010−12−31
+1646−50
+329.7 - 331.4
+2011−12−31
+1148−671
+302.4 - 304.0
+2011−01−01
+1511−55, 1646−50
+328.0 - 329.7
+2012−01−01
+1148−671
+300.7 - 302.3
+2011−01−02
+1511−55, 1646−50
+326.3 - 328.0
+2012−01−02
+1352−63, 1148−671
+307.5 - 309.1
+2011−01−04
+1511−55
+324.6 - 326.3
+2012−01−03
+1148−671, 1352−63
+294.3 - 295.4
+2011−01−05
+1511−55
+322.9 - 324.6
+2012−01−04
+1352−63
+312.6 - 314.2
+2011−01−06
+1511−55
+321.2 - 322.9
+2012−01−05
+1148−671, 1352−63
+301.5 - 326.2
+2011−01−07
+1511−55
+319.5 - 321.2
+2012−01−07
+1352−63
+310.1 - 310.9
+2011−12−20
+1646−50, 1511−55
+333.1 - 334.7
+Flagging parameters were determined manually for each block
+of observations and written to a master configuration file that was
+applied automatically when the pipeline was run. After an initial
+flagging of the primary and secondary calibrators, bandpass and
+flux calibrations were performed. For blocks with two secondary
+calibrators, the solutions in phase and amplitude were estimated for
+each calibrator and then combined to produce a global calibration
+table. A second pass of flagging was then performed on the calibrated
+data before performing a second and final calibration. This was to
+ensure that the final calibration was performed on properly flagged
+data. The final global flagging and calibration tables produced were
+then applied to the science data and split into individual pointings.
+4 IMAGING
+The calibrated visibilities of individual fields were imaged by iter-
+atively cleaning down to the maximum residual peak flux (MRF:
+see Section 4.1) using multi-frequency synthesis (mfs; Sault & Con-
+way 1999). This is implemented in MIRIAD using invert with the
+’mfs’, ’sdb’ options and the multi-frequency CLEAN mfclean tasks.
+Multi-frequency imaging accounts for the spectral variation across
+the observation bandwidth of 2-GHz. Thus, the resulting image con-
+sists of the normal flux component and the flux times the spectral
+component (I𝛼), where 𝑆𝜈 ∝ 𝜈𝛼.
+The dirty images were created using a robust weighting scheme
+(Briggs 1995) of 0.5 robustness and gridded to an image pixel size of
+0.6′′, ∼, which is about one third of the minor axis of the synthesized
+beam. The robust weighting scheme provides a trade-off between
+uniform and natural weighting, which is a trade-off between resolu-
+tion and sensitivity. The choice of 0.5 allows an improved sensitivity
+without sacrificing the resolution. To ensure uniform resolution, we
+have forced a restoring circular Gaussian beam with an FWHM of
+2.5′′ using restor (see Section 5.2). The residuals were not corrected
+for the change in resolution, so care should be taken when summing
+values below the maximum residual flux (see 4.1).
+4.1 Maximum Residual Flux
+The maximum residual flux (MRF) is an important parameter in the
+deconvolution process and should be carefully chosen. CLEANing
+deeper than necessary would result in many artefacts in the form
+of faint spurious sources and under-estimated peak fluxes and flux
+densities of real sources. This is due to the CLEAN algorithm clean-
+ing noise spikes and re-distributing flux from point sources to noise
+peaks and is known as the clean bias effect (White et al. 1997; Condon
+et al. 1998). This effect is more pronounced for sources with lower
+signal-to-noise ratio (SNR), given that the amount of distributed flux
+is independent of the flux density of the source. According to Pran-
+doni et al. (2000), the clean bias effect can be reduced by setting
+the MRF above the noise level. However, a shallow CLEAN, where
+the MRF is too far above the noise level, will create residual images
+dominated by sidelobes. Therefore, it is important that MRF be care-
+fully chosen, especially for a blind survey such as CORNISH. Unlike
+for the CORNISH-North survey the mfclean task does not employ
+windowing. Equation (1) is an expression for the MRF, where fmrf
+is a constant multiplicative factor and rms is the root mean square
+noise level.
+MRF ≃ fmrf × rms
+(1)
+Given that the clean bias is also dependent on the rms noise level,
+we have determined the rms in Equation (1) by imaging the central
+portion of each field’s dirty map in Stokes V. Stokes V maps usually
+have very few sources or no sources, since there are few circularly
+polarized sources at 5.5-GHz (Roberts et al. 1975; Homan & Lister
+2006). Hence, the Stokes V maps are dominated by thermal noise.
+In order to determine an average and appropriate fmrf value for the
+CORNISH-South data, we followed the same procedure as in Paper II.
+Artificial point sources and sources with simple morphologies were
+introduced into the uv data of an empty field and the field was imaged
+with fmrf values from 0.5 to 5, using our imaging pipeline. The
+flux densities of these artificial point sources were then measured
+and compared with the original flux densities. Figure 2 shows the
+resulting plot of the fraction of recovered flux density against the
+fmrf values. The range of fmrf values resulted in a > 98 per cent
+recovered flux density. Below fmrf=1.5 the field is over CLEANed
+MNRAS 000, 1–22 (2022)
+
+Data processing and catalogue.
+5
+1
+2
+3
+4
+5
+fmrf
+0.98
+0.99
+1.00
+Percentage Recovered Flux (%)
+Figure 2. Fraction of recovered flux density vs. fmrf for an artificial source of
+50 mJy that is introduced into an empty field (115050.00-612353.14). Shaded
+grey region represents the ratio of the rms noise level to the recovered flux
+density.
+with a >40 per cent decrease in the rms noise level, compared to the
+Stokes V map.
+To further constrain the choice of an average fmrf value, for each
+iteration, the residual images of the sources with simple morpholo-
+gies were inspected for sidelobe structures. At fmrf=1.5, the Stokes
+I map is CLEANed deeply enough to remove sidelobes and recover
+> 99 per cent of the flux density. Based on these tests, an average
+value of fmrf = 1.5 was used in the imaging process. For fields with
+imaging artefacts as a result of very bright sources (> 1Jy) and/or
+extended sources, we have re-imaged with higher values of fmrf and
+manually CLEANed, where necessary. We discuss the procedure for
+estimating the effect of the clean bias on the CORNISH-South data
+in Section 5.4.
+4.2 Imaging Extended Sources
+The shortest baseline of the ATCA 6A configuration is ∼337 m,
+which corresponds to a spatial scale of ∼40′′. Nevertheless, the
+uv-plane at these short baselines is sparsely sampled and the de-
+convolution algorithm models extended emission as a series of delta
+functions. Thus, the algorithm struggles to re-construct extended
+emission due to the difficulty in interpolating the sparsely sampled
+plane. This is reflected in the image as scattered flux, which results
+in high rms noise and imaging artefacts, especially in regions with
+bright and large structures.
+To estimate the maximum recoverable angular size above which
+the deconvolution begins to fail or struggles to recover the flux density
+of extended sources, we injected a 1 Jy artificial Gaussian source into
+the uv data of an empty field. Holding the flux density constant, the
+FWHM was increased from 3 to 30′′ in 2′′ steps. The data were
+imaged with our imaging pipeline and then the injected Gaussian
+properties were measured for each iteration using the AEGEAN
+source fitting algorithm (see Section 6.1). Figure 3 shows the fraction
+of recovered flux density as a function of the injected Gaussian
+FWHM (top panel) and a plot of the measured sizes against the
+injected FWHM (bottom panel).
+For sizes above 17′′, the source was fitted with two Gaussians
+by AEGEAN; the FWHM plotted in Figure 3 (bottom panel) is the
+Gaussian with the larger 𝜃maj (red line). Our pipeline combines the
+5
+10
+15
+20
+25
+30
+0.2
+0.4
+0.6
+0.8
+1.0
+Fractional Recovered Flux
+5
+10
+15
+20
+25
+30
+Injected FWHM (arcsec)
+5
+10
+15
+20
+25
+30
+Fitted FWHM (arcsec)
+Figure 3. Top panel: Fractional recovered flux density as a function of FWHM
+for an artificial Gaussian source of 1 Jy. Bottom panel: Recovered sizes as a
+function of the injected FWHM. The red line traces the Gaussian fits while
+the black line after 17′′ traces the polygon fit. The black dashed line is a fit
+to the Gaussian measurements. The transition from Gaussian fitted sizes to
+polygon sizes happens between 17′′ and 20′′ (see text for details). Both plots
+share same x-axis.
+multiple Gaussians into a polygon and measures the geometric mean
+as the angular size (see Section 6.1), which is traced by the black
+line. The maximum FWHM of the fitted Gaussian is ∼ 17′′ (red line)
+as illustrated in Figure 3 (bottom panel). However, by switching to
+a polygon that encloses the emission, we can recover up to ∼ 24′′
+(black line). Above 17′′, the corresponding flux density is reduced
+by ≥ 40 per cent as the flux density steeply drops off. For the real
+data, we expect that the reduction in flux density and limited size
+of an extended source should be influenced by morphology as well
+as rms noise level. Compared to the northern counterpart with an
+estimated maximum size of ∼ 14′′ (Paper II), we expect to see more
+extended sources in the CORNISH-South catalogue. Most structures
+that are less than 15′′should give reasonable estimates of the angular
+MNRAS 000, 1–22 (2022)
+
+6
+T. Irabor et al.
+size and the integrated flux density is expected to be within a factor
+of 2 of the true value.
+4.3 Mosaicking
+Individually imaged fields were linearly mosaicked onto 20′ × 20′
+grid tiles, using linmos, a MIRIAD task13 that overlap by 1′). The
+tiles are arranged in equatorial coordinates (J2000) and 1825 tiles
+were needed to cover the survey region, extending to the edges of
+the survey. Tiles covering the edges for |b| > 1◦ may not be full; i.e
+less than 20′, and some sources at the edges of the survey region
+may have poor image fidelity due to poor uv-coverage. Wide-band
+primary beam correction, which is the inverse of the primary beam
+response as a function of the radius and frequency, was performed by
+linmos for each field before linearly mosaicking them to form a tile.
+Linmos uses the 𝛼I plane and takes into account the OTF scanning
+during the primary beam correction. In order to improve the accuracy
+of the primary beam correction, the ‘bw’ option in linmos was used
+to specify the bandwidth of the images (2-GHz).
+Linear mosaicking is performed in linmos using the standard mo-
+saic equation by minimizing the rms noise (Sault et al. 1996). In
+order to properly account for the geometry and avoid interpolation
+problems during mosaicking, the overlapping fields were put on the
+same pixel grid using the ‘offset’ key in invert. If this is not taken into
+account, the position of sources in overlapping tiles will be altered
+because linmos does not automatically account for the geometric
+correction during linear mosaicking.
+5 DATA QUALITY
+5.1 Calibrators
+The six secondary calibrators along the Galactic plane are shown in
+Figure 1. For observation days with two secondary calibrators, cali-
+bration was done separately and then the solutions were combined.
+Before imaging the science data, the calibrators were imaged. This
+was to inspect the images for artefacts, jets or anything else that could
+a��ect the data. All the secondary calibrators as shown in Figure 4
+are point sources with no jets or any other structure ≥ 5𝜎 within the
+field.
+B1934-638 was the preferred primary flux calibrator with a flux
+density of 4.95 Jy at 5.5-GHz. An additional backup flux calibra-
+tor (B0823-500) with a flux density of 2.93 Jy at 5.5-GHz was
+also observed at the beginning of each day’s observation. B0823-
+500 was used for flux calibration when the B1934-638 data were
+bad or when it could not be observed due to time constraints e.g.
+for blocks 28 and 35 (2011-12-30 and 2012-01-07). The flux den-
+sities of the secondary calibrators from the ATCA calibrator man-
+ual8 at 5.5-GHz are 1.064 ± 0.007 Jy (1352-63), 1.557 ± 0.007 Jy
+(1148-671), 2.338 ± 0.009 Jy (1511-55), 2.063 ± 0.018 Jy (1646-
+50), 1.172 ± 0.006 Jy (1714-397) and 1.73 ± 0.01 Jy (1729-37). Be-
+cause the calibrators are standard calibrators, mfcal uses the appro-
+priate flux density variation with frequency during calibration.
+With the available data, the flux densities of the secondary cali-
+brators were measured after flagging and calibration. Figure 5 shows
+the deviation in percentage from the median flux densities of the six
+13 https://www.atnf.csiro.au/computing/software/miriad/
+doc/linmos.html
+8 https://www.narrabri.atnf.csiro.au/calibrators/
+calibrator_database.html
+secondary calibrators over the 35 days of observations. Each point in
+Figure 5 represents a single day’s observations. Different calibrators
+were used for the different epochs with the exception of 1511-55 that
+overlaps both epochs. The secondary calibrators show a percentage
+flux density deviation from the mean flux density that is less than 10
+per cent and a standard deviation of 4.0 per cent. Based on the scatter
+in the flux density deviation of the secondary calibrators in Figure 5,
+we have adopted a 10 per cent calibration error for the CORNISH-
+South data. The mean positional accuracy of all six calibrators9 is
+0.1′′.
+5.2 Synthesized Beam
+Figure 6 (bottom right) shows the distribution of the unconstrained
+major and minor axes of the dirty beam. The major axis has a median
+of ∼2.5′′ and extends up to 5.5′′. The minor axis distribution shows a
+narrower distribution with a peak about 1.8′′. For uniformity across
+the survey region, the median value of the major axis distribution
+(2.5′′) was chosen as the size of a circular restoring beam. This
+results in a super-resolution (major axis/restoring beam of 2.5′′)
+distribution presented in Figure 6 (bottom middle). Ninety percent
+(90 per cent) of the fields have super-resolution that is less than 1.3.
+We also plan to release the calibrated uv dataset so that users can re-
+image fields with whichever beam they require, as well as convolving
+the residuals with their chosen beam.
+The resulting elongation of the synthesized beam before super-
+resolution, i.e. the ratio of the major to the minor axis, is shown in
+Figure 6 (bottom right). Ninety-six percent (96 per cent) of the fields
+have elongation less than 2 with a peak at ∼1.4. This means that there
+are a few fields (<4 per cent) where the major axis is 2 or more times
+greater than the minor axis. The variation of the beam’s major axis
+across the survey region is presented in Figure 6. The elongation of
+the synthesized beam of an East-West array like ATCA is a function
+of the declination of the field, hence the greater elongation for l >
+344◦ for more equatorial declinations. This will also explain the fields
+with major axis > 3′′ (4′′ > 𝜃maj > 3′′) seen within the longitude
+region >344◦ (Figure 6 : Top panel). Fields with higher major axes
+(> 3.5′′) in Figure 6 fall within the longitude 333◦ to 335◦ region.
+Figure 6 (top panel) also highlights the epochs. Epoch II shows a
+less elongated beam, with major axes lower than 3′′ for ∼93 per cent
+of the fields. The fields with higher major axes within the longitude
+333◦ to 335◦ region are seen to come from epoch II. This is due to
+the poor uv-coverage resulting from less than the typical 12 scans for
+the block observed on 2011-12-20. However, only a few fields were
+affected, making up ∼6 per cent of the epoch II data.
+5.3 Sensitivity/Root Mean Square (rms) noise Level
+The rms noise level achieved across the survey region in CLEANed
+Stokes I maps is shown in Figure 7. The noise level is fairly uniform,
+having a mean of 0.11 mJy beam−1. The rms noise level around a few
+very bright source clusters is particularly high as expected. The noisy
+region between longitude 333◦ and 335◦ is seen to reflect the poor
+uv-coverage seen in Figure 6. Figure 7 (bottom panel) further shows
+a distribution of the rms noise with an elongated tail that corresponds
+to regions with poor uv-coverages. The two peaks correspond to the
+different epochs, where the second peak is dominated by epoch I
+data. The noise level of epoch II data (hatched grey shaded region) is
+better than that of epoch I (hatched red shaded region). Given that the
+9 https://www.narrabri.atnf.csiro.au/calibrators/calupdate.html
+MNRAS 000, 1–22 (2022)
+
+Data processing and catalogue.
+7
+13h55m50s48s
+46s
+44s
+-63°26'15"
+30"
+45"
+27'00"
+RA (J2000)
+DEC (J2000)
+1352-63
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Jy/beam
+11h51m18s 15s
+12s
+09s
+-67°27'45"
+28'00"
+15"
+30"
+RA (J2000)
+DEC (J2000)
+1148-671
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+Jy/beam
+15h15m16s 14s
+12s
+10s
+-55°59'15"
+30"
+45"
+-56°00'00"
+RA (J2000)
+DEC (J2000)
+1511-55
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+1.6
+Jy/beam
+16h50m18s
+16s
+14s
+-50°44'30"
+45"
+45'00"
+15"
+RA (J2000)
+DEC (J2000)
+1646-50
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Jy/beam
+17h17m41s40s
+39s
+38s
+37s
+-39°48'30"
+45"
+49'00"
+15"
+RA (J2000)
+DEC (J2000)
+1714-397
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Jy/beam
+17h33m17s16s
+15s
+14s
+13s
+-37°22'15"
+30"
+45"
+23'00"
+RA (J2000)
+DEC (J2000)
+1729-37
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+1.4
+Jy/beam
+Figure 4. Images of the six secondary calibrators 1352-63 (1.064 ± 0.007 Jy), 1148-671 (1.557 ± 0.007 Jy), 1511-55 (2.338 ± 0.009 Jy), 1646-50 (2.063 ± 0.018
+Jy), 1714-397 (1.172 ± 0.006 Jy), 1729-37 (1.73 ± 0.01 Jy). The quoted flux densities are from the ATCA calibrator manual. The images are all 1′ by 1′ with
+the size of the beam shown at the bottom left.
+0
+5
+10
+15
+20
+25
+30
+35
+Day/Block Number
+10.0
+7.5
+5.0
+2.5
+0.0
+2.5
+5.0
+7.5
+10.0
+Percentage deviation from median flux (%)
+1714-397
+1729-37
+1646-50
+1511-55
+1352-63
+1148-671
+Figure 5. Percentage deviation from the median flux density vs. block/day
+number for the six secondary calibrators. Each point represents measurements
+from a block’s observations.
+two epochs are separated by eleven months, the intervening series of
+system maintenance and tests 10 could have improved the efficiency
+and overall performance of the array. The region of epoch II data
+with rms noise > 0.11 mJy beam−1 corresponds to fields where there
+were fewer than the average twelve scans (see Section 2).
+5.4 Clean Bias
+To estimate the effect of the clean bias by CLEANing down to the
+MRF (see Equation 1), point sources of random flux densities be-
+tween 1 and 15 mJy were injected into the uv data of six empty
+tiles chosen from the two epochs. The positions of the sources were
+chosen to fall about the centres of individual fields, away from any
+source. The fields were imaged and mosaicked using the imaging
+10 https://www.narrabri.atnf.csiro.au/observing/schedules/
+2011OctSem/CA.pdf
+pipeline. The flux densities and sizes were then measured from the
+mosaicked tiles at the injected positions, using our aperture photom-
+etry pipeline. This was to avoid flux density bias caused by thermal
+flux fluctuations (Franzen et al. 2015). This procedure was repeated
+ten times to get a better average estimate of the clean bias.
+Measured flux densities were subtracted from the injected flux
+densities to get an estimate of the clean bias effect. The median clean
+bias from averaging the measurements is estimated to be 0.14 mJy
+and was not different across the epochs. This value is about half the
+clean bias estimated for the CORNISH-North catalogue of 0.33 mJy
+(Paper II). For ≥7𝜎 sources, this effect is < 13 per cent. Thus, we
+conclude that the clean bias will not significantly degrade the quality
+of the flux densities compared to the statistical and absolute flux
+density calibration uncertainty of 10 per cent.
+6 CATALOGUE
+6.1 Source Finding and Characterization
+The automated source finding and source characterization part of
+our pipeline utilizes the AEGEAN software package11 (Hancock
+et al. 2012, 2018). AEGEAN uses the flood-fill algorithm, where
+two thresholds (𝜎s: seeding threshold, 𝜎f: flooding threshold) are
+defined, such that 𝜎s ⩾ 𝜎f. The seeding threshold is used to seed
+an island, while the flooding threshold is used to grow the island
+(see Hancock et al. 2012). Detected pixels are then grouped into
+contiguous islands and characterized by fitting with one or more
+overlapping 2D Gaussians.
+We have used the background and noise estimation function, Bane
+11 https://github.com/PaulHancock/Aegean/wiki/
+Quick-Start-Guide
+MNRAS 000, 1–22 (2022)
+
+8
+T. Irabor et al.
+300
+310
+320
+330
+340
+350
+Galactic Longitude (deg)
+1
+2
+3
+4
+5
+Major Axis/Super Resolution/Beam Elongation
+Super Resolution
+Major Axis
+Elongation
+Epoch I
+Epoch II
+0
+1
+2
+3
+4
+5
+Major/Minor Axis (arcsec)
+0
+500
+1000
+1500
+2000
+2500
+Number of fields
+Major
+Minor
+0.0
+0.5
+1.0
+1.5
+2.0
+Beam Super Resolution (arcsec)
+0
+500
+1000
+1500
+2000
+Number of fields
+0
+1
+2
+3
+Beam Elongation
+0
+500
+1000
+1500
+2000
+2500
+Number of fields
+Figure 6. Top panel: Galactic longitude distribution of the major axis and beam elongation before super-resolution is shown. The synthesized beam of an
+East-West array like ATCA is a function of the declination of the field, hence the greater major axis for l > 344◦. Additionally, the super-resolution distribution
+is presented. Bottom panel: Major and minor axes distribution of the imaged fields with a median major (𝜃maj) of 2.5′′ and median minor axis (𝜃min) of 1.8′′
+(bottom left). Forcing a restoring beam size of 2.5′′ (FWHM) will result in the distribution of the beam super-resolution shown in the middle panel (bottom). 66
+per cent of the fields have super-resolution that is greater than 1. The beam elongation is the ratio of the major to the minor axis (bottom right) . 96 per cent of
+the fields have elongation less than 2.
+(Hancock et al. 2012) to compute the background and rms noise for
+each tile. Bane uses a grid algorithm that estimates the rms noise
+(𝜎BANE) and background level within a sliding box of a defined size
+that is centered on a grid point. The pixel values within the box
+are then subjected to sigma clipping (3𝜎), which reduces any effect
+source pixels may introduce (Hancock et al. 2012; see also Bertin &
+Arnouts 1996). Given that radio images do not have very complicated
+backgrounds, we do not expect the noise properties across a 20′ × 20′
+tile to change much. Therefore, we used the default boxcar size
+of 30𝜃bm (𝜃bm: synthesized beam size) for the CORNISH-South
+data, which is about 75′′. This size has been demonstrated to be
+optimized for the completeness and reliability of compact sources
+(Huynh et al. 2012). For source finding we defined a 4.5𝜎𝑠 seeding
+threshold and 4.0𝜎f flooding threshold to create an initial catalogue
+of the CORNISH-South data.
+6.2 Quality Control
+6.2.1 Elimination of Duplicate Sources
+With a 60′′ overlap of the tiles, sources closer to the edges of the
+tiles were detected more than once. However, because overlapping
+regions are formed from the same fields, the difference in position
+would be a fraction of the synthesized beam. This will also affect
+the peak flux, depending on the local rms noise. An extended source
+fitted with multiple Gaussians could have different parameters from
+one tile to another because it is closer to the edge on one tile and
+fully imaged on another tile.
+To
+eliminate
+such
+duplicated
+sources,
+we
+searched
+for
+sources with similar positions, < 2.0′′ and similar peak flux,
+Peakmin/Peakmax > 0.7. In addition to both conditions, the distance
+of each duplicated source, relative to the centre of the tile, was cal-
+culated the source closer to the centre of a tile was retained, over the
+ones closer to the edges.
+MNRAS 000, 1–22 (2022)
+
+Data processing and catalogue.
+9
+300
+310
+320
+330
+340
+350
+Galactic Longitude (deg)
+0.075
+0.100
+0.125
+0.150
+0.175
+0.200
+0.225
+0.250
+RMS noise level (mJy/beam)
+Epoch I
+Epoch II
+0.08
+0.10
+0.12
+0.15
+0.17
+0.20
+0.23
+0.25
+RMS noise level (mJy/beam)
+100
+101
+102
+103
+Number of fields
+Epoch I
+Epoch II
+Figure 7. Top panel: The variation of the rms noise in Stokes I maps across
+Galactic longitude. Bottom panel: Distribution of the rms noise in Stokes
+I maps (grey region) measured within an aperture size of 3′. The hatched
+regions represent the rms noise from the epoch I (red hatched region) and II
+(black hatched region. The mean rms noise is ∼ 0.11 mJy beam−1. The SIQR
+of epoch I is 0.003 and that of epoch II is 0.004.
+6.2.2 Elimination of Spurious Sources
+The choice of a cut-off threshold for a catalogue, in terms of the SNR,
+is a trade-off between completeness and reliability. A low threshold
+catalogue, e.g. 3𝜎, will result in more real sources but with many
+unreal sources as well, while a high threshold will result in a highly
+reliable catalogue but miss real sources with low SNR. In order to
+determine an appropriate cut-off threshold for the highly reliable
+CORNISH-South catalogue, we attempted to estimate the number
+of spurious sources at a given threshold. Based on the analysis in
+Paper II (see Hopkins et al. 2002), 15 tiles were selected to represent
+all 1825 tiles. These tiles were chosen such that there were no sources
+with very bright side lobes and contained only point sources and fairly
+extended sources.
+To estimate the number of spurious sources as a function of SNR,
+i.e. the ratio of peak flux to rms noise level, the tiles were inverted by
+multiplying the pixel values in the tiles by -1. A seeding threshold of
+4.5𝜎s was then used to search for sources on both sets of tiles (normal
+and inverted tiles). To account for an average estimate of the number
+of sources across the survey region, the detections from the 15 tiles
+were multiplied by 122 (1825/15). Figure 8 shows a cumulative
+histogram of the detected sources before and after inversion (the
+later being obviously all spurious), as a function of the SNR. The
+rms noise level used for the SNR is measured in an annulus with a
+5′′ gap from the source aperture (𝜎a: see Section 6.2.3). Detections
+below 5𝜎𝑎 are dominated by spurious sources (> 90 per cent) as
+indicated by the grey shaded region. Spurious detections fall off
+steeply compared to real sources above 5𝜎a and then fall off to 1 in
+the 15 tiles between 6.0𝜎a and 6.5𝜎a.
+Following the analysis in Paper II, if the population of detected
+sources is assumed to be governed by Gaussian statistics, then the
+fraction, f(𝜎), of the population that falls within a given detection
+threshold can be expressed as
+f(𝜎) = 1 − errf(𝜎/
+√
+2) ,
+(2)
+where errf(𝜎) is the Gaussian error function, given by
+errf(𝜎) =
+1
+√𝜋
+∫ 𝜎
+−𝜎
+e−t2dt =
+2
+√𝜋
+∫ 𝜎
+0
+e−t2dt.
+(3)
+A plot of f(𝜎) is presented in Figure 8, assuming the total number of
+possible detections equals the number of beams within the CORNISH
+survey region (2.02 × 108 beams). With this assumption, the total
+number of spurious sources is underestimated (blue dashed line).
+However, the number of sources can be allowed to be a free parameter,
+resulting in a fit represented by the black line. The black line appears
+to predict the number of spurious sources at 4.5𝜎a but falls off rather
+too steeply, compared to the number of spurious sources. Fitting
+f(𝜎) to bins higher than 5𝜎a and adjusting the width of the Gaussian
+(𝜎 = 0.8𝜎gauss) results in a better fit (green dashed line) and predicts
+the number of spurious sources to be less than 10 at 7𝜎a. Figure 8
+shows that the fraction of spurious sources decreases from 25 per cent
+to 5 per cent to 1 per cent at 5𝜎, 5.5𝜎 and 6𝜎, respectively. Based
+on this analysis, the cut-off threshold for a reliable CORNISH-South
+catalogue to be accepted is set at 7𝜎a.
+6.2.3 Gaussian Sources
+For compact sources fitted with a single Gaussian by AEGEAN,
+the integrated flux densities we adopt those given by AEGEAN.
+However, for the rms noise we re-measure this in an annulus around
+an aperture (𝜎a). The source aperture was defined by an elliptical
+aperture which extends to 3𝜎 of the Gaussian major (𝜃maj) and
+minor (𝜃min) axes. An annulus with the same shape as the source
+aperture of width 15′′, offset at 5′′ from the source aperture, was then
+defined to measure the rms noise and background level. The choice
+of an annulus offset of 5′′ allows an estimation of the background
+around the immediate locale of the source. An annulus of width 15′′
+provides a statistically large area in pixels over which to compute the
+background and rms noise level, compared to the synthesized beam
+area of 19.7 pixels. This is more local than 𝜎BANE and is consistent
+with what was used for the CORNISH-North survey. Paper II provides
+equations and further details on the aperture photometry (see also
+Paper IV).
+6.2.4 Extended Non-Gaussian Sources
+Extended non-Gaussian sources were automatically detected by
+searching for contiguous islands with more than one overlapping
+Gaussian. A single optimal 2D polygon was then defined using the
+convex hull algorithm to trace the outer outline of the Gaussians, en-
+closing the emission. The assumption is that overlapping Gaussians
+MNRAS 000, 1–22 (2022)
+
+10
+T. Irabor et al.
+Figure 8. Number of sources as a function of signal-to-noise ratio for 15 real
+and inverted tiles. The hatched grey region represents spurious sources, while
+the hatched blue region represents real sources. The blue line is a fit to the
+spurious sources assuming the total number of possible detections equals the
+number of beams within the CORNISH survey region (2.02 × 108 beams).
+The black line is the fit to the number of spurious sources by allowing it to be a
+free parameter and adjusting the curve to fit. The green line represents a fit to
+binshigher than 5𝜎a and adjustingthe widthofthe Gaussian(𝜎 = 0.8𝜎gauss).
+trace a single extended source. The extent of the generated 2D poly-
+gon is strongly affected by the extent of the individual Gaussians.
+Thus, before generating the polygons, there was the need to make
+sure the catalogue is clean (i.e. free from sidelobes), otherwise the
+generated polygon may be over-estimated, stretched by the sidelobes.
+Additionally, some extended sources that were not properly imaged
+may appear as individual Gaussians, spreading over an area. In such
+cases, manual intervention was needed to trace the outline of the
+real emission. Such cases account for ∼ 10 per cent of non-Gaussian
+sources.
+Given the defined polygons, new intensity weighted centres (𝛼0,
+𝛿0) and diameters were then determined. The diameter of the 2D
+polygon, defined by n-sides and n-vertices, was calculated by deter-
+mining the radius of each vertex to the intensity weighted centre and
+then estimating the geometric mean of the radii and multiplying by a
+factor of 2. Figure 9 shows an example of two sources that were first
+fitted by multiple Gaussians and the subsequently generated poly-
+gons. For these extended sources, aperture photometry was used to
+measure the source properties within the defined polygon that en-
+closes the emission and the rms noise level within the annulus (see
+Paper II and Paper IV). The geometric mean of the 2D polygon is
+𝜃E = 2 � 𝑛√𝑟1����2...𝑟𝑛
+�, where r1, r2...rn are the radii of the vertices.
+Having removed duplicates, spurious sources, and sources < 7𝜎a
+(see Section 6.2.2), we then used visual inspection to further elimi-
+nate artefacts due to side-lobes that are close to very bright sources.
+This was aided by comparison with the GLIMPSE (Churchwell et al.
+2009) data. As we are primarily interested in a complete sample of
+UCHII regions, radio sources in areas with artefacts that have clear
+IR counterparts were retained. Often these artefacts are caused by
+bright H II regions themselves and we expect other H II regions to be
+present in such clustered star forming regions. After these elimina-
+tions, the final CORNISH-South catalogue has a total of 4701 high
+quality sources above 7𝜎a.
+6.3 Measurements and Uncertainties
+In order to create a uniform catalogue that is similar to the CORNISH-
+North catalogue, we have used the same sets of equations given
+in Paper II to estimate the properties and associated errors of the
+sources (also see Condon 1997). For the well-defined and unresolved
+sources, defined by a single Gaussian fit, the AEGEAN Gaussian
+fit measurements are the catalogued properties. However, for the
+catalogued rms noise level, we have re-measured within an annulus
+around the source for both extended and non-extended sources. This
+was to create a CORNISH (North and South) catalogue with uniform
+noise measurements, given that we have implemented sigma clipping
+to remove sources within the annulus (see Paper II).
+The integrated flux density and associated error are given by
+S =
+A𝜋
+4ln(2)
+𝜃maj𝜃min
+𝜃2
+bm
+(4)
+and
+𝜎2
+S
+S2 ≈
+𝜎2
+A
+A2 +
+𝜃2
+bm
+𝜃maj𝜃min
+������
+𝜎2(𝜃maj)
+𝜃2
+maj
++ 𝜎2(𝜃min)
+𝜃2
+min
+������
+,
+(5)
+where A is the peak amplitude, 𝜃min is the minor axis, 𝜃maj is the
+major axis and 𝜃bm is the synthesized beam size. 𝜎𝜃maj and 𝜎𝜃min
+are the errors on the Gaussian fits. The catalogued angular size and
+associated error is the geometric mean of the major and minor axes,
+which can be calculated from
+𝜃mean =
+√︃
+𝜃maj𝜃min
+(6)
+and
+𝜎(𝜃mean) = 𝜃mean
+2
+�
+�
+�𝜎2(𝜃maj)
+𝜃2
+maj
++ 𝜎2(𝜃min)
+𝜃2
+min
+(7)
+The integrated flux density of the polygonal sources, measured
+using aperture photometry, is given by
+S =
+�Nsrc
+∑︁
+i=1
+Ai − NsrcB
+� �
+abm ,
+(8)
+where �Nsrc
+i=1 Ai is the total flux density in a given aperture over Nsrc
+pixels, B is the median background flux and abm is the beam area
+(19.66 pixels). The associated error is given by
+𝜎2
+S =
+�
+𝜎(
+∑︁
+Ai)2 +
+𝜋N2src𝜎2g
+2Nsky
+� �
+a2
+bm ,
+(9)
+where 𝜎2g is the variance and 𝑁𝑠𝑘𝑦 is the number of pixels in the
+annulus. For CORNISH-North, the angular sizes for the polygonal
+sources were intensity-weighted diameters and were given by
+dw =
+Nsrc
+∑︁
+i=1
+riAi
+� Nsrc
+∑︁
+i=1
+Ai ,
+(10)
+where dw is the intensity-weighted diameter and �Nsrc
+i=1 Ai is the
+sum of the flux within the defined source aperture. This works well
+for simple extended sources. However, the sizes of very extended
+and double-lobed sources could be under-estimated by ≥50 per
+cent, in some cases. Figure 10 shows a comparison of the intensity-
+weighted diameters and the geometric mean diameters for the polyg-
+onal sources (see Section 6.2.4). The intensity-weighted diameters
+MNRAS 000, 1–22 (2022)
+
+105
+real sources
+spurious sources
+Cumulative detections (x 122)
+104
+103
+102
+101
+100
+5
+6
+8
+9
+7
+10
+Signal-to-noise threshold (o)Data processing and catalogue.
+11
+13h42m48s 45s
+42s
+39s
+36s
+-61°52'20"
+40"
+53'00"
+20"
+RA (J2000)
+DEC (J2000)
+G308.9310+0.3917
+13h42m48s 45s
+42s
+39s
+36s
+-61°52'20"
+40"
+53'00"
+20"
+RA (J2000)
+DEC (J2000)
+G308.9310+0.3917
+14h57m40s
+36s
+32s
+28s
+-58°11'30"
+12'00"
+30"
+13'00"
+RA (J2000)
+DEC (J2000)
+G318.9310+0.6956
+14h57m40s
+36s
+32s
+28s
+-58°11'30"
+12'00"
+30"
+13'00"
+RA (J2000)
+DEC (J2000)
+G318.9310+0.6956
+Figure 9. Examples of generated polygons for extended sources fitted with multiple Gaussian by the AEGEAN source finder. The fitted Gaussians are overlaid
+(left panel), while the defined polygon is shown (right panel). The top source (radio galaxy) is an example of a source that shows the centre source as a Gaussian
+and the lobes as non-Gaussian sources. The source at the bottom is an example of a double lobe HII region.
+are consistently smaller compared to the geometric mean diameters.
+The best-fit line to the scatter is given by: y = 0.44x − 0.41. Based on
+this, the catalogued size for a polygonal source is the geometric mean.
+This can be useful in interpreting and re-scaling the CORNISH-North
+sizes for extended sources.
+6.4 Completeness
+It is important to demonstrate the completeness of our 5.5-GHz cata-
+logue as a function of flux density. In order to quantify this, artificial
+point sources were injected into the calibrated uv-data of nine fairly
+empty tiles having no imaging artefacts, fairly homogenous noise
+distribution and from both epoch I and epoch II. The flux densities of
+the artificial point sources were chosen to be in the range of 0.2 mJy
+(∼ 2𝜎) to 4 mJy (∼ 40𝜎). The positions and flux densities were ran-
+domly assigned, while avoiding positions of real sources and making
+sure that the artificial sources do not overlap. In total, 5000 sources
+were injected into the tiles. The tiles were imaged, and the proper-
+ties of the injected sources were measured with the same pipeline
+that produced the catalogue. This procedure was repeated 10 times
+and then the average values of the measured properties over the 10
+iterations were compared to the injected properties.
+Figure 11 shows the completeness level measured by the percent-
+age of detected sources as a function of their injected flux densities.
+The mean completeness level (percentage) is also shown as a black
+5
+10
+15
+20
+25
+30
+35
+40
+Geometric Mean Diameter (arcsec)
+0
+5
+10
+15
+20
+25
+Intensity Weighted Diameter (arcsec)
+one-to-one line
+best fit line: y = 0.44x
+0.41
+Figure 10. A plot of the intensity-weighted diameter against the geometric
+mean diameter for the polygonal sources. The one-to-one line (black) shows
+that the geometric mean diameter is consistently larger than the intensity
+weighted diameter. The best-fit line (green) is given by y = 0.44x − 0.41.
+line with the ’+’ symbol. The percentage completeness from the
+graph shows > 90 per cent for 1.5 mJy and essentially 100 per cent
+for > 3 mJy. The completeness for <0.3 mJy is 0 per cent because
+it is below the seeding threshold of 4.5𝜎s. Table 4 shows the noise
+MNRAS 000, 1–22 (2022)
+
+12
+T. Irabor et al.
+0.5
+1.0
+1.5
+2.0
+2.5
+3.0
+3.5
+4.0
+Flux Density (mJy)
+0
+20
+40
+60
+80
+100
+Recovered sources (%)
+Tile 7
+Tile 62
+Tile 81
+Tile 109
+Tile 1614
+Tile 1655
+Tile 1731
+Tile 1733
+Tile 1752
+Mean
+Figure 11. Percentage completeness as a function of flux density for artificial
+point sources injected into 9 representative tiles. Completeness is the number
+of extracted sources divided by number of injected sources at binned flux
+densities. The black line with the ’+’ symbol shows the mean completeness.
+Table 4. Completeness level of the 5.5-GHz CORNISH-South data across 9
+representative tiles at 50 per cent and 90 per cent.
+Tile
+Epoch
+rms
+50 per cent
+90 per cent
+(mJy beam−1)
+mJy
+mJy
+7
+II
+0.09
+0.63
+0.85
+62
+II
+0.09
+0.43
+0.64
+81
+II
+0.09
+0.45
+0.68
+109
+II
+0.10
+0.63
+0.88
+1614
+I
+0.19
+0.81
+1.38
+1655
+I
+0.14
+0.80
+1.30
+1731
+I
+0.12
+0.72
+1.27
+1733
+I
+0.12
+0.51
+1.02
+1752
+I
+0.16
+0.42
+1.14
+Mean
+I & II
+0.12
+0.60
+1.09
+level and completeness for the individual tiles at 50 per cent and
+90 per cent. Tiles from epoch I with higher noise levels show lower
+completeness level compared to tiles from epoch II. The stated rms
+noise level is an average across each tile and so the completeness
+will be affected by local rms noise surrounding a given source. At
+1.4 mJy (∼7𝜎), the percentage completeness is about 90 per cent for
+the worst case (Tile 1614). Based on the mean completeness level, for
+point sources, the CORNISH-South data is 90 per cent complete at
+1.1 mJy. The completeness will be worse around very bright sources,
+but as can be seen from Figure 7, this represents less than 0.3 per
+cent of the total area of the survey.
+6.5 Catalogue Ensemble Properties
+Figures 12 to 14 present the distribution of the ensemble physi-
+cal properties of the CORNISH-South sources. We identified 4701
+sources above the 7𝜎 limit, of which the properties of 608 are mea-
+sured with a polygon.
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Angular Size (arcsec)
+100
+101
+102
+103
+Number of Sources
+Figure 12. Angular size distribution of the 7𝜎 CORNISH-South sources.
+6.5.1 Angular Size
+The catalogued angular size for both the Gaussian and non-Gaussian
+sources is the geometric mean (see section 6.3). Based on the mean
+error of the angular sizes, which is ∼ 0.3′′ and the size of the restoring
+beam (2.5′′), resolved sources are defined as sources with angular
+sizes > 2.8′′ for the CORNISH-South catalogue. The angular size
+distribution in Figure 12 is dominated by unresolved sources (66.3
+per cent) and accounts for the obvious peak at ∼ 2.5′′. Resolved
+sources (1584) account for 33.6 per cent of the catalogue, of which
+38 per cent are polygonal sources (608). The distribution of resolved
+sources is fairly flat out to 30′′ after a steep drop from 2.5′′ to 5′′.
+This is consistent with the maximum recoverable size (see Figure 3).
+As characteristic of all interferometric observations, very extended
+emission will not be properly imaged due to missing information on
+large scale structures, limited by the shortest baseline in the array.
+Thus, caution should be applied in interpreting the angular sizes
+and flux densities of very extended sources (> 17′′). This is also
+demonstrated in Section 4.2.
+6.5.2 Galactic Latitude and Longitude Distributions
+Figures 13a and 13b show the distributions of the CORNISH-South
+sources in Galactic latitude and longitude, respectively. The cover-
+age of the CORNISH-South survey is complete within the |b| ⩽ 1.0
+region as shown in Figure 13a. The distributions are similar com-
+pared to the Galactic distributions of the CORNISH-North catalogue
+(Paper II).
+The latitude and longitude distributions of the resolved sources
+correspond to the Galactic region traced by high-mass star formation
+(Urquhart et al. 2011, 2009). Known star formation complexes G333
+and G338.398+00.164, can be seen at l = 333◦ and l = 338◦, respec-
+tively (Urquhart et al. 2013a; Urquhart et al. 2013b). Based on the
+Galactic distribution of the resolved sources, they are expected to be
+dominated by HII regions that are concentrated towards the Galactic
+mid-plane (Urquhart et al. 2013a, Paper III).
+6.5.3 Flux Density Distribution
+The integrated flux density and peak flux distribution in Figure
+14 shows similar distributions compared to the CORNISH-North
+sources (Paper II). The flux density distribution peaks at ∼1 mJy,
+MNRAS 000, 1–22 (2022)
+
+Data processing and catalogue.
+13
+(a) Galactic latitude distribution.
+1.0
+0.5
+0.0
+0.5
+1.0
+Galactic Latitude  (deg)
+0
+50
+100
+150
+200
+250
+300
+Number of Sources
+All Sources
+Extended Sources
+(b) Galactic longitude distribution with a bin size of 2◦.
+300
+310
+320
+330
+340
+350
+Galactic Longitude  (deg)
+0
+50
+100
+150
+200
+Number of Sources
+All Sources
+Extended Sources
+Figure 13. Galactic latitude (13a) and longitude (13b) distributions of the
+7𝜎 CORNISH-South sources.
+below which the number of sources drops off due to increasing in-
+completeness (see Section 6.4). For the resolved/extended sources,
+the flux density distribution peaks at ∼ 3 mJy and gently falls off,
+extending up to 104 mJy. Compared to the CORNISH-North, we
+have picked up more faint sources as expected due to the sensitivity
+being two times better.
+6.6 Source classification and Example Sources
+Initial classification of the CORNISH-South sources has utilized
+the availability of comparable high-resolution and high-sensitivity
+surveys (GLIMPSE, VVV, VPHAS, Hi-Gal and ATLASGAL) of
+the Galactic plane. As with the CORNISH-North survey, one of
+us (MGH) has visually inspected the multi-wavelength images of
+each source on the CORNISH-South website. Examples of multi-
+wavelength images of the main different types of sources are shown
+in Figure A1. Using experience gained from classification of sources
+in the RMS (Red MSX Source) survey (Lumsden et al. 2013), the
+following visual classification criteria were used.
+HII regions have strong, and usually extended, mid-IR, far-IR
+and sub-millimetre counterpart emission. The morphology of the
+IR emission is usually irregular and complex, as well as often be-
+100
+101
+102
+103
+104
+Integrated Flux Density (mJy)
+100
+101
+102
+Number of Sources
+All Sources
+Extended Sources
+Figure 14. Distributions of the integrated flux density of the 7𝜎 CORNISH-
+South catalogue. Resolved/extended sources are represented by the hatched
+regions. Figure 19 in Paper II shows the distribution of the CORNISH-North.
+ing part of a clustered environment. Their radio emission is usually
+fairly strong and when resolved can have a cometary, shell or ir-
+regular morphology. The mid-IR emission, dominated by polycyclic
+aromatic hydrocarbons (PAH) emission, arises from just outside the
+radio emitting region and often reflects the same morphology (Hoare
+et al. 2007). An initial sub-classification on angular size has been used
+in the catalogue , pending distance information. All H II regions less
+than 5′′ in size have been labelled as UCHIIs. This corresponds to
+the typical 0.1 pc size of UCHIIs if they were at a typical distance
+of ∼ 4 kpc for the more nearby part of the population of UCHIIs
+(Paper III). The larger ones were mostly labelled as H II regions if
+the radio emission was clearly identifiable as part of a single source
+across the radio and IR bands. If the radio emission was due to over-
+resolution of a much larger, multiple and complex source in the IR
+then it was labelled as a diffuse H II region. A few HII regions were
+hidden behind large amounts of dust extinction at 8𝜇m and were
+labelled as IR-Dark H II regions.
+PNe also have strong mid-IR counterparts due to dust and PAH
+emission (Smith & McLean 2008; Guzman-Ramirez et al. 2014; Cox
+et al. 2016), but are much fainter at far-IR wavelengths than H II re-
+gions, and are usually undetected in sub-millimetre plane surveys.
+The SEDs of PNe generally peak at ∼ 24𝜇m but some young and
+dense PNe have their peaks extending up to 70𝜇m and beyond (see
+Paper IV; Anderson et al. 2012; Urquhart et al. 2013a). A few of
+the bipolar, Type I, PNe can have more far-IR and sub-millimetre
+emission, but are still significantly weaker than H II regions. Mor-
+phologically they are much simpler than H II regions and are isolated
+rather than being in clustered, complex environments.
+Radio stars are point sources in every waveband they are detected
+in. They are also isolated sources in the field. Depending on the type
+of radio star they can either have blue or red colours in the optical
+and IR. Very red radio stars like dusty symbiotic stars are difficult to
+distinguish from unresolved PN without further information (Irabor
+et al. 2018).
+Due to the sensitivity of the CORNISH-South survey the radio
+emission from a few known massive young stellar objects (MYSOs)
+was detected. They share all the IR characteristics of H II regions, but
+have very weak radio emission compared to HII regions in general
+and are unresolved or jet-like (Purser et al. 2016). It can be difficult
+MNRAS 000, 1–22 (2022)
+
+14
+T. Irabor et al.
+0
+200
+400
+600
+800
+1000
+Number of Sources
+All HII Region
+IR Quiet
+MYSO
+Other
+PN
+Radio Galaxy
+Radio Star
+Source Type
+Figure 15. Distribution of classified CORNISH-South sources. Fifty-percent
+has been classified so far. All HII regions sum up to 530, of which 257 are
+UCHII regions. Unclassified sources add up to ∼ 2300.
+to distinguish weak, unresolved UCHIIs powered by B3 stars from
+MYSOs as they have similar radio luminosities (Purser et al. 2016).
+Most extra-galactic radio sources are undetected in corresponding
+optical, IR and sub-millimetre Galactic plane surveys used here and
+therefore they are straightforward to identify. In the radio, they are
+usually unresolved single sources and we classify them as IR quiet
+sources in the catalogue. A small number of these may be radio stars
+that are so distant or obscured as to remain undetected in the optical
+and IR surveys. A significant number of extra-galactic sources show
+the classic double radio source morphology, and these are classified
+as “Radio Galaxy (Lobe)” in the catalogue. In some cases, the core
+of the radio galaxy is also seen and is classified as such. If both
+lobes, or a lobe and a core, or both lobes and a core are visible, but
+part of the same extended source in the catalogue they are referred
+to as “Radio Galaxy (Both)”. It is possible that some normal, star-
+forming galaxies, as opposed to AGN, are detected in CORNISH-
+South. Some radio sources had very faint, resolved counterparts in the
+mid-IR, where it was not clear if they are very distant PNe or galaxies.
+Further studies will be required to rule out a PN classification.
+A few sources did not fit in to any of the above categories, either
+because they are known sources of unusual type, or their origin is
+currently unknown. These were classified as “Other”.
+At the time of publication, all sources with detectable flux at 8 𝜇m
+seen within the same aperture used for radio fluxes have been clas-
+sified. This should account for the vast majority of Galactic sources.
+Figure 15 shows the distribution of classifions of these sources. Re-
+solved sources with infrared counterparts are dominated by PNe and
+HII regions. We find more UCHII regions compared to the northern
+counterpart and have also identified six known MYSOs. With a sen-
+sitivity of 0.11 mJy beam−1, we have also detected PNe with lower
+radio flux density compared to CORNISH-North. We expect that the
+vast majority of the sources that remain to be visually classified will
+be extra-galactic and under the IR Quiet or Radio Galaxy categories.
+The latest classifications will be those on the CORNISH-South web-
+site.
+6.6.1 Catalogue Format
+An excerpt of the catalogue is presented in Table 6 and the columns
+are arranged in the following format: Column (1) - CORNISH Source
+name ((l + b)); Columns (2) and (3) - right ascension (𝛼) and declina-
+tion (𝛿) in (J2000) with their associated errors in brackets; Column
+(4) – Peak flux and associated error in mJy beam−1; Column (5)
+- integrated flux density and associated error in mJy. Because the
+clean bias is close to the rms noise level and within the errors, the
+flux densities were not corrected for the clean bias effect. Column
+(6) - Angular size and associated error in arcsec; Column (7) - Gaus-
+sian FWHM major axis and error in arcsec; Column (8) - Gaussian
+FWHM minor axis and error in arcsec; Column (9) - position angle
+(E of N) of elliptical Gaussian; Column (10) - local rms noise level,
+(𝜎a), in mJy beam−1 as measured in the annulus described in Section
+6.2.3; Column (11) - Signal to noise ratio of the source given by Peak
+flux divided by 𝜎a; Column (12) - Type of source tells if the source is
+Gaussian fitted (G) or non-Gaussian, in which case a polygon (P) was
+drawn around the source; Column (12) - This column indicates the
+classification of the source. Gaussian sources have both the Gaussian
+fitted sizes and geometric mean sizes in the final catalogue. The final
+version of the full table is made available on the CORNISH website
+(http://cornish.leeds.ac.uk/public/index.php).
+6.7 Astrometry and Flux Density Quality Check
+6.7.1 GLIMPSE
+In order to check the astrometry of the CORNISH-South catalogue,
+we cross-matched a sub-set of the CORNISH-South sources with the
+GLIMPSE point source catalogue. To avoid mis-matches and multi-
+ple matches, the CORNISH-South catalogue was limited to classified
+sources, excluding extended HII regions, radio-galaxies and infrared
+quiet sources. Additionally, sources with angular size > 3′′ were ex-
+cluded. The cross-match returned 218 sources and Figure 16 shows
+the distributions of the offsets in right ascension (𝛼) and declination
+(𝛿).
+The distribution of the angular offsets in Figure 16 (top left) peaks
+at ∼0.4′′ and steeply falls to 1.5′′ before continuing gently out to 3′′.
+The offset distribution in 𝛼 is tightly peaked at about 0′′ compared
+to the distribution in 𝛿 that has double peaks at 0.5 and -0.2′′. The
+spread in 𝛿 offset can be attributed to the distribution of the intrinsic
+beam along the major axis seen in the epoch I data (see Figure ??).
+6.7.2 The Red MSX Source (RMS) 6 cm ATCA survey
+Targeted radio continuum observations were conducted to identify
+UCHII regions and PNe as part of the Red MSX Source (RMS) survey
+(Lumsden et al. 2013). The observations were carried out with the
+ATCA within the 235◦ < l < 350◦ region at 3.6 and 6 cm (Urquhart
+et al. 2007b). Table 5 compares the observational parameters of
+the RMS 4.8-GHz (6 cm) and the CORNISH-South 5.5-GHz. Both
+surveys have similar observational properties but the CORNISH-
+South observing bandwidth of 2-GHz provides better image fidelity
+compared to the 128 MHz of the RMS survey.
+For further checks on the CORNISH-South astrometry and
+flux densities, the CORNISH-South 5.5-GHz catalogue was cross-
+matched with the RMS 4.8-GHz catalogue (Urquhart et al. 2007a)
+within a 5′′ radius. For a one-to-one match, the CORNISH-South
+catalogue was limited to only sources that have been visually classi-
+fied, excluding diffuse HII regions and radio-galaxies. Figure 17 (top
+left) shows the distribution of the angular separation between the
+CORNISH-South and RMS for 186 radio sources. The distribution
+shows a tight correlation with a sharp peak at 0.3′′ and then a steep
+fall to ∼ 1.5′′ before gently falling off to ∼ 4.3′′.
+Seventy-five per cent of the cross-matched sources fall within 1.5′′
+and 94 per cent fall within 3′′. Compared to the distribution of the
+offset in 𝛿, the offset in 𝛼 shows a narrower distribution that is strongly
+peaked at 0′′. Similar offset distribution in 𝛿 is seen in the CORNISH-
+South-GLIMPSE cross-match that is attributed to the spread in the
+intrinsic major axis distribution. The mean offset in 𝛼 and 𝛿 is 0.1′′
+MNRAS 000, 1–22 (2022)
+
+Data processing and catalogue.
+15
+3
+2
+1
+0
+1
+2
+3
+d  (arcsec)
+0
+5
+10
+15
+20
+25
+30
+35
+40
+Number of Sources
+-3
+-2
+-1
+0
+1
+2
+3
+d  (arcsec)
+-3
+-2
+-1
+0
+1
+2
+3
+d  (arcsec)
+3
+2
+1
+0
+1
+2
+3
+d  (arcsec)
+0
+5
+10
+15
+20
+Number of Sources
+Figure 16. CORNISH-South and GLIMPSE cross-matched sources (218) within 3′′. Top left: The angular offset between cross-matched sources. Top right:
+Offset distribution in 𝛼 (arcsec). Bottom left: Offset distribution in 𝛿 (arcsec). Bottom right: Scatter plot of offsets in 𝛼 against 𝛿. The cross symbol indicates
+the mean in 𝛼 (0.02 ± 0.04′′) and 𝛿 (0.19 ± 0.04′′ ). The error is the standard error on the mean.
+Table 5. Comparison of the RMS 6 cm (4.8-GHz) and the CORNISH-South
+observation parameters.
+Parameters
+CORNISH-South
+RMS
+Rest frequency (GHz)
+5.5
+4.8
+Array
+6A
+6C/6D
+Bandwidth (GHz)
+2
+0.128
+Synthesised beam
+2.5′′
+2.5′′
+Typical image rms (mJy beam−1)
+0.11
+0.27
+Image pixel size
+0.6′′
+0.6′′
+and 0.2′′, respectively. Based on this, and the CORNISH-South-
+GLIMPSE cross-match, we adopt a positional accuracy of 0.22 ±
+0.11′′ for the CORNISH-South catalogue. This is also in line with the
+mean positional accuracy of the secondary calibrators (see Section
+5.1).
+In Figure 18, the flux densities of the cross-matched sources are
+compared. A one-to-one line (black line) shows that the CORNISH-
+South 5.5-GHz flux densities are higher than the RMS 4.8-GHz flux
+densities on average. The few sources where the RMS flux densities
+are higher are found to be HII regions with larger angular sizes. The
+two surveys used different ATCA configurations (see Table 5), which
+are sensitive to different angular sizes. Compared to the shortest
+baselines of the 6C (153 m) and 6D (77 m) configurations used for
+the RMS survey (Urquhart et al. 2007b), the shortest baseline of the
+CORNISH-South 6A configuration (337 m) makes it less sensitive to
+extended structures. Hence, for optically thin and angularly large HII
+regions, the RMS flux densities are expected to be higher, as seen in
+Figure 18. However, the better uv-coverage of the CORNISH-South
+recovers more extended emission on some sources.
+A best-fit line (dotted green line) predicts ∼ 24 per cent flux density
+increase for the CORNISH-South counterparts that is more than the
+typical calibration error of 10 per cent. According to Urquhart et al.
+(2007b), the distribution of spectral indices, (𝛼, where S𝜈 ∝ 𝜈𝛼)
+between the 3.6 and 6 cm data is slightly skewed towards positive
+indices, which suggests optically thick sources in their sample. This
+provides a possible explanation for the compact sources where the
+5.5 GHz CORNISH-South flux densities are higher than the 4.8 GHz
+RMS ones.
+7 CONCLUSION AND FUTURE WORK
+The CORNISH program has successfully mapped the southern
+Galactic plane at radio wavelengths with unprecedented resolution
+and sensitivity. We have presented radio continuum ATCA data at
+5.5-GHz, covering the 295◦ < l < 350◦; |b| ≤ 1◦ region of the south-
+ern Galactic plane. The resolution of 2.5′′ and noise level of 0.11
+mJy beam−1 deliver radio data matched to the existing high reso-
+lution, multi-wavelength surveys GLIMPSE, VVV and VPHAS+ of
+the southern Galactic plane.
+Utilizing the MIRIAD program for data reduction and AEGEAN
+source finding algorithm, we have identified 4701 sources above 7𝜎.
+Data arising from fields with poor uv-coverage make up to only 2 per
+cent of the data set. In addition to several measures undertaken to
+ensure data quality, visual inspection has also been used to exclude
+artefacts and hence the data are highly reliable. The survey has a 90%
+completeness level at a flux density of 1.1 mJy. The measured proper-
+ties show distributions that are similar to that of the CORNISH-North
+catalogue presented in Paper II.
+All sources with potential counterparts at 8𝜇m (GLIMPSE) have
+been visually classified. This corresponds to 43 per cent of the
+sources. We have identified 530 HII regions, of which 257 are
+UCHIIs. Additionally, we identified 287 PNe and 79 radio stars.
+The vast majority of the remaining unclassified sources without in-
+frared counterparts are expected to be extra-galactic. With the sen-
+sitivity of the CORNISH-South survey being two times better than
+the CORNISH-North counterpart, sources with lower flux densities
+have been detected, including a few MYSOs. A detailed analysis of
+the statistical properties of individual catalogues will be presented
+in future papers. The CORNISH-South survey also carried out si-
+multaneous observations of the entire field at 9-GHz, which will be
+presented in a separate paper. The 9-GHz data provide higher spatial
+resolution at reduced sensitivity and will enable an examination of
+spectral indices of the sources.
+The CORNISH-South data are particularly important in the char-
+acterization of compact ionized sources towards the Galactic mid-
+plane, as no radio survey has previously covered the southern Galactic
+plane in such resolution and sensitivity. Previous surveys of the south-
+ern Galactic plane were limited in resolution and coverage, especially
+within the |b| < 1◦ region, hence not suitable for studies of compact
+ionized regions. New radio facilities such as the, ASKAP12(Hotan
+12 https://www.atnf.csiro.au/projects/askap/index.html
+MNRAS 000, 1–22 (2022)
+
+16
+T. Irabor et al.
+4
+2
+0
+2
+4
+d  (arcsec)
+0
+5
+10
+15
+20
+25
+30
+35
+Number of Sources
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+d  (arcsec)
+-5
+-4
+-3
+-2
+-1
+0
+1
+2
+3
+4
+5
+d  (arcsec)
+4
+2
+0
+2
+4
+d  (arcsec)
+0.0
+2.5
+5.0
+7.5
+10.0
+12.5
+15.0
+17.5
+Number of Sources
+Figure 17. CORNISH-South and RMS 4.8-GHz cross-matched radio sources (186) within 5′′. Top left: The angular offset between cross-matched sources. Top
+right: Offset distribution in 𝛼 (arcsec). Bottom left: Offset distribution in 𝛿 (arcsec). Bottom right: Scatter plot of offsets in 𝛼 against 𝛿. The cross symbol
+indicates the mean in d𝛼 (0.12 ± 0.07′′) and d𝛿 (0.21 ± 0.08′′). The mean error for the CORNISH-South is 8 mJy.
+100
+101
+102
+103
+104
+CORNISH 5.5 GHz (mJy)
+100
+101
+102
+103
+104
+RMS 4.8 GHz (mJy)
+10
+15
+Maximum RMS 4.8 GHz size (arcsec)
+Figure 18. A plot of the 5.5-GHz CORNISH-South flux densities against
+the RMS 4.8-GHz (Urquhart et al. 2007b) flux densities. The black line is a
+one-to-one line (y = x) and the dotted green line is the best fit line defined
+by log(y) = (0.99 ± 0.03)log(x) − 0.11 ± 0.05. The error on the fitted line
+is the standard deviation.
+et al. 2014), MeerKAT13(Jonas & MeerKAT Team 2016) and ul-
+timately the SKA14(Braun et al. 2019) are exploring the southern
+Galactic plane at greater depth. The CORNISH-South catalogue is
+well positioned as a very useful resource to characterise the popu-
+lation of radio sources seen in the MEERKAT L-band survey (8′′
+resolution and 10𝜇Jy/beam noise level; Goedhart et al., In prepara-
+tion.) and also for follow-up observations with ALMA15. It will also
+be useful in combination with existing multi-wavelength surveys of
+the southern Galactic plane to address key questions in stellar for-
+mation and evolution. This will allow multi-wavelength exploration
+and statistical studies of compact ionized regions, which can then be
+compared with population synthesis models. To date, the CORNISH
+13 https://www.sarao.ac.za/gallery/meerkat/
+14 https://www.skatelescope.org/
+15 https://www.almaobservatory.org/en/home/
+program has delivered the most sensitive and unbiased compact ra-
+dio source catalogue towards the southern Galactic mid-plane. Data
+products in the form of FITS images of individual sources and cata-
+logues are available on the CORNISH-South Website 16.
+With deeper and larger radio surveys on the way, it is also im-
+portant that we look at building and improving existing machine-
+learning models for image classification (e.g. Adhiambo et al., in
+preparation). Having used multi-wavelength visual classification, the
+CORNISH sources provide a good training and validation set for
+machine-learning models that will allow automated classification of
+radio sources in these upcoming larger and deeper surveys.
+16 https://cornish-south.leeds.ac.uk/
+MNRAS 000, 1–22 (2022)
+
+Data processing and catalogue.
+17
+Table 6. An excerpt of the 5.5-GHz CORNISH-South catalogue. The full version of the catalogue is available online. Measurement errors are in parentheses. The flux densities have not been corrected for the clean
+bias effect.
+Source Name
+𝛼 (h m s)
+𝛿 (◦ ′ ′′)
+Peak
+S5.5−GHz
+𝜃s
+𝜃maj
+𝜃min
+PA
+rms
+Sigma
+Type
+Class
+(l+b)
+(J2000)
+(J2000)
+(mJy
+beam−1)
+mJy
+(′′)
+(′′)
+(′′)
+(deg)
+mJy beam−1
+G295.1757−0.5744 11:43:38.84 (0.65)
+-62:25:17.5 (0.44)
+4.12 (0.21)
+58.13 (7.20)
+20.84 (0.14)
+0.0
+0.0
+0.0
+0.12
+23
+P
+HII Region
+G296.5033−0.2695 11:55:25.27 (0.02)
+-62:26:24.6 (0.02)
+10.92 (1.22)
+15.06 (1.87)
+8.43 (0.05)
+0.0
+0.0
+0.0
+0.10
+72
+P
+PN
+G297.3943−0.6347 12:02:23.29 (0.96)
+-62:58:42.8 (1.28)
+0.94 (0.02)
+7.09 (2.61)
+16.67(0.34)
+0.0
+0.0
+0.0
+0.10
+9.9
+P
+HII Region
+G298.3434+0.1466
+12:11:42.23 (0.21)
+-62:22:14.9 (0.35)
+0.74 (0.01)
+1.30 (0.81)
+7.18(0.22)
+0.0
+0.0
+0.0
+0.08
+9.8
+P
+Radio Galaxy
+G298.8382−0.3388 12:15:20.73 (0.03)
+-62:55:25.4 (0.03)
+11.29 (1.30)
+35.34 (2.89)
+11.29 (0.06)
+0.0
+0.0
+0.0
+0.17
+68
+P
+HII Region
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+...
+G349.9207+0.6682
+17:16:31.30 (0.02)
+-36:59:40.29 (0.02)
+13.16 (0.26)
+13.92 (0.61)
+2.57 (0.06)
+2.62 (0.05)
+2.52(0.05)
+0.73
+0.17
+75
+G
+PN
+G349.9260+0.0811
+17:18:56.62 (0.07)
+-37:19:44.7 (0.11)
+2.61 (0.08)
+17.17 (0.54)
+6.42 (0.01)
+7.75 (0.01)
+5.31 (0.01)
+-0.71
+0.14
+21
+G
+HII Region
+G350.0290−0.4950 17:21:37.33 (0.16)
+-37:34:25.77 (0.18)
+2.64 (0.42)
+2.58 (0.92)
+2.48 (0.44)
+2.72 (0.41)
+2.25 (0.37)
+-7.17
+0.35
+7.4
+G
+Radio Galaxy
+G350.0920+0.2309
+17:18:48.44 (0.01)
+-37:06:25.28 (0.01)
+89.33 (0.46)
+96.32 (1.08)
+2.60 (0.02)
+2.68 (0.01)
+2.52 (0.01)
+-8.31
+0.27
+330
+G
+PN
+G350.1237−0.5563 17:22:08.88 (0.06)
+-37:31:50.61 (0.07)
+3.68 (0.2)
+3.27 (0.49)
+2.36 (0.18)
+2.74 (0.17)
+2.03 (0.15)
+-0.78
+0.12
+29
+G
+IR Quiet
+MNRAS 000, 1–22 (2022)
+
+18
+T. Irabor et al.
+ACKNOWLEDGEMENTS
+TI acknowledges the support of the Science and Technology Facilities
+Council of the United Kingdom (STFC) through grant ST/P00041X/1
+and the OSAPND scholarship (Nigeria). The work done by PFG was
+carried out in part at the Jet Propulsion Laboratory, which is operated
+by the California Institute of Technology under a contract with the
+National Aeronautics and Space Administration (80NM0018D0004).
+JM acknowledges financial support from the State Agency for Re-
+search of the Spanish Ministry of Science and Innovation under grant
+PID2019-105510GB-C32. JMP acknowledge financial support from
+the State Agency for Research of the Spanish Ministry of Science
+and Innovation under grant PID2019-105510GB-C31 and through
+the Unit of Excellence María de Maeztu 2020-2023 award to the In-
+stitute of Cosmos Sciences (CEX2019-000918-M). G.A.F acknowl-
+edges support from the Collaborative Research Centre 956, funded by
+the Deutsche Forschungsgemeinschaft (DFG) project ID 184018867.
+The Australia Telescope Compact Array (ATCA) is part of the
+Australia Telescope National Facility which is funded by the Aus-
+tralian Government for operation as a National Facility managed by
+CSIRO. We acknowledge the Gomeroi people as the traditional own-
+ers of the Observatory site. This work made use of Montage, aplpy
+and astropy python libraries in the batch processing of FITS files.
+DATA AVAILABILITY
+All data (images and catalogues) are described in the text and avail-
+able on the cornish website at https://cornish-south.leeds.
+ac.uk/ to download in csv (catalogues), uv files and FITS (images)
+formats.
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+
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+APPENDIX A: EXAMPLE SOURCES IN THE
+CORNISH-SOUTH CATALOGUE THAT HAVE BEEN
+CLASSIFIED.
+MNRAS 000, 1–22 (2022)
+
+20
+T. Irabor et al.
+Figure A1. From the left: 5.5-GHz radio (CORNISH), 3-colour GLIMPSE, 70𝜇m image (Hi-Gal) and ATLASGAL 850𝜇m image . From top to bottom: Example HII region, UCHII region, PN, radio-star,
+radio-galaxy and infrared-quiet sources. CORNISH radio images are 80′′ by 80′′; GLIMPSE images are 100′′ by 100′′; HI-GAL and ATLASGAL images are 180′′ by 180′′. The red polygons/gaussian overlays are
+that of the 5.5 GHz CORNISH sizes. Images are available online.
+MNRAS 000, 1–22 (2022)
+
+G311.4260+0.5967
+-61°05'20"
+Dec (J2000)
+40"
+4
+mjy/beam
+2
+.00.90
+0
+20"
+Hll Region
+14h02m39s
+36s
+33s
+RA (J2000)-61°04'30"
+05'00"
+30"
+.00.90
+30"
+07'00"
+14h02m48s
+42s
+36s
+30s-61°04'30"
+05'00"
+30"
+..00.90
+30"-
+07'00"
+14h02m48s
+42s
+365
+305
+24s-53°47'40"
+G327.8480+0.0179
+-10
+8
+48'00"
+6
+Dec (2000)
+mJy/beam
+20"
+4
+2
+40"
+0
+UCHIl Region
+15h53m32s
+30s
+28s
+26s
+RA(J2000)53°47'00"
+30"
+48'00"
+0
+30"
+49'00"
+30"
+15h53m35s
+30s
+25s
+20s-53°47'00"
+30"
+48'00"
+0
+30"
+49'00"
+30"
+15h53m35s
+30s
+25s
+20s-37°53'00"
+G349.3700-0.1105
+8
+20"
+Dec (J2000)
+mjy/beam
+4
+40"
+54'00"
+0
+Planetray Nebula
+17h18m10s
+08s
+06s
+04s
+RA (J2000)-37°52'30"
+53'00"
+30"
+54'00"
+30"
+55'00"
+17h18m12s
+08s
+04s
+s00-37°52'30"
+53'00"
+30"-
+54'00"
+30"
+55'00"
+17h18m12s
+08s
+04s
+00sData processing and catalogue.
+21
+Figure A2. Continuation from Figure A1. Images are available online.
+MNRAS 000, 1–22 (2022)
+
+G336.5160+0.0656
+47°48'00"
+2.0
+1.5
+20"
+Dec (J2000)
+mJy/beam
+1.0
+40"
+0.5
+0.0
+49'00"
+Radio Star
+-0.5
+16h33m18s
+16s
+14s
+12s
+RA(J2000)47°47'30"
+48'00"
+30"
+49'00"
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+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+110 
+ 
+Szabolcs Nagy – Gergő Hajdú 
+ 
+The relationship between content marketing and the traditional  
+marketing communication tools 
+ 
+Digitalization is making a significant impact on marketing. New marketing approaches and tools 
+are emerging which are not always clearly categorised.  This article seeks to investigate the 
+relationship between one of the novel marketing tools, content marketing, and the five elements of 
+the traditional marketing communication mix. Based on an extensive literature review, this paper 
+analyses the main differences and similarities between them. This article aims to generate a debate 
+on the status of content marketing. According to the authors' opinion, content marketing can be 
+considered as the sixth marketing communication mix element. However, further research is 
+needed to fill in the existing knowledge gap. 
+Keywords: content marketing, trends, advertising, sales promotion, direct marketing, personal 
+selling, public relations 
+JEL: M31, M37 
+ 
+https://doi.org/10.32976/stratfuz.2021.25  
+ 
+Introduction 
+ 
+Digitalization and the ongoing information technology revolution pose remarkable possibilities 
+and challenges for marketing (Piskóti, 2018). Due to digitalization, consumer behaviour is 
+constantly changing. Consumers’ stimulus threshold is increasing because of the greater exposure 
+to information (Kotler et al., 2017). At the same time, smart devices are becoming increasingly 
+dominant (Nagy, 2017). E-commerce (Nagy, 2016), and the various social networks are becoming 
+popular (Sethi, 2018). These trends accelerate the emergence of new methods and trends in 
+marketing (Nagy, 2020). It is advisable for marketers to understand how those new methods and 
+tools work since they help to reach out to consumers to influence their behaviour. However, the 
+lack of advanced information technology in Hungary poses some problems in this process 
+(Kamaraonline, 2018). 
+Marketing communication tools can often be divided into two main groups. Traditional and digital 
+solutions can be distinguished. However, according to Kotler et al. (2017), the two categories have 
+recently been merging. Content marketing is essentially a digital solution having some offline 
+features as well. The significance of content marketing is supported by Kotler et al. (2017), who 
+found - referring to the research findings of Content Marketing Institute and the MarketingProfs - 
+that 76% of the B2C companies and 88% of the B2B companies used content marketing in North 
+America. Furthermore, B2C companies spent 32% of their marketing budgets on content 
+marketing, while B2B companies spent 28%. 57% of the B2C companies increased their content 
+marketing budget by at least 1%, while 29% of them did not change the budget (Brenner 2019, 
+based on Content Marketing Institute 2019). The companies mainly increased their content 
+marketing budgets in the following areas: content production (56%), content marketing personnel 
+(37%), paid distribution of content (36%), content marketing technology (29%), and content 
+marketing outsourcing (29%) (Murton Beets 2018). These facts also underline the importance of 
+content marketing in today’s digital world.  
+If we accept that traditional and digital solutions have been merging (Kotler et al., 2017), it means 
+that the traditional classification of marketing communication tools should be revised. The 
+communication tools should rather be classified according to their functions and operating 
+mechanisms than according to the type of technological solutions. From this perspective, content 
+marketing (CM) is a new approach to marketing communication and a novel marketing 
+communication tool that can be combined with traditional marketing tools. Therefore, the present 
+paper seeks to investigate the relationship between content marketing and the five, traditional 
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+ 
+111 
+ 
+marketing communication tools to generate discussion if content marketing is the sixth element 
+of the revised marketing communication mix. 
+ 
+Literature review 
+ 
+Content marketing definition, functions, and spending 
+ 
+Content marketing is the creation and distribution of relevant, timely, and valid content (Wang et 
+al., 2017). Its primary purpose is to create customer trust and value (Repoviener, 2017). Content 
+marketing may have entertaining or educational functions (Duc Le 2016; Lindström and Jörnéus, 
+2016). Content marketing can be effectively used both in B2C and B2B markets (Iankova et al. 
+2019). According to Kotler et al. (2017), the content can serve brand-building or sales promotion 
+purposes. According to Moutsos (2019), 55% of the companies were capable of generating sales 
+and income, and 53% of them were capable of increasing their existing customers' loyalty through 
+content marketing in 2018. So, content marketing can be used to generate income and sales, and 
+also, to increase customers' loyalty. 
+ 
+Content types and formats 
+ 
+Content marketing may appear in various formats based on the type of content. It could be audio 
+and/or visual content (videos, live streaming, webinars); written digital content (articles, blogs, 
+ebooks), images (infographics, photos, GIFs, charts), in-person content (events, presentations, 
+workshops); audio-only digital content (podcasts, audiobooks), and written print content 
+(magazines, books, brochures).  Figure 1. shows the different types of content and how B2B 
+marketers changed their use of content types/formats. Figure 2. shows the very same trends in 
+B2C markets.  
+    
+ 
+Figure 1. The change of use of content types/format in B2B markets 
+Source: Own compilation based on Murton Beets, 2018   
+ 
+As Figure 1 illustrates, in B2B markets, the use of audio/visual content; written digital content and 
+images became more popular, while the use of written print content significantly decreased 
+compared to the other types. The same trends can be seen in the B2C markets (Figure 2). The only 
+slight difference between the two markets is in the use of audio-only digital content, which 
+significantly dropped in the B2C market. 
+ 
+64%
+61%
+56%
+41%
+38%
+27%
+32%
+33%
+40%
+52%
+54%
+56%
+4%
+6%
+4%
+7%
+8%
+17%
+Audio/Visual Content
+Written Digital Content
+Images
+In-Person Content
+Audio-only Digital Content
+Written Print Content
+Increased
+Remained the same
+Decreased
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+112 
+ 
+ 
+Figure 2. The change of use of content types/format in B2C markets 
+Source: Own compilation based on Murton Beets, 2018   
+ 
+In practice, various types of content can be used to reach out to consumers. As far as the type of 
+content concerned, e-mail campaign is the most popular one, used by 87% of the companies 
+(Murton Beets 2018).  However, the following content types are also frequently used (values in 
+brackets show the percentage of companies using the given content type): educative content 
+(77%), actions calling for the next step (62%), events involving personal interactions (61%), 
+telling stories (45%), offers (27%) and community building involving the public (23%). Trends 
+and forecasts are less popular, only 5% of the companies used them (Murton Beets 2018).  
+ 
+The goals of content marketing  
+ 
+Content marketing helps to achieve several goals. The goal of content marketing is to gain 
+customers (Barker, 2017) and to build customer relationships (Pažėraitė and Repovienė, 2018). 
+Content marketing can very effectively be used to create brand awareness, educate audiences, 
+generate demand/leads, and build credibility/trust (Figure 3.). 
+Also, content marketing is an effective tool for nurturing subscribers/audience/leads; driving 
+attendance to one or more in-person events, building loyalty with existing clients, and supporting 
+the launch of a new product. It can even be used to achieve sales/revenue generation and build a 
+subscribed audience. Figure 3. presents the possible goals companies managed to successfully 
+achieve by using content marketing. 
+ 
+69%
+64%
+63%
+37%
+30%
+27%
+25%
+31%
+30%
+51%
+48%
+63%
+6%
+5%
+7%
+12%
+22%
+10%
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+100%
+Audio/Visual Content
+Written Digital Content
+Images
+In-Person Content
+Audio-only Digital Content
+Written Print Content
+Increased
+Remained the same
+Decreased
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+ 
+113 
+ 
+ 
+Figure 3. Content marketing goals 
+Source: Own compilation based on Content Marketing Institute (2019).  
+Note: Goals enterprise marketers have achieved by using content marketing successfully   
+ 
+ 
+Research methodology 
+ 
+This paper seeks to generate a debate on the current state of content marketing, and it aims to 
+create a base for future quantitative research. It synthesizes the relevant literature to analyze the 
+relationship between content marketing and the traditional marketing communication tools. It 
+makes an attempt to distinguish content marketing from the other elements in marketing 
+communication mix, which are advertising, sales promotion (SP), public relations (PR), personal 
+selling, and direct marketing (DM). In the following section, based on extensive literature review, 
+the five traditional marketing communication tools are compared to content marketing to reveal 
+the similarities and differences between them regarding the type, purpose, standardization, time 
+span and reach of communication and the target groups. 
+ 
+Research findings and discussion 
+ 
+The relationship between advertising and content marketing 
+ 
+Advertising is the most prominent element of the traditional communication mix. According to 
+Horváth and Bauer (2013) advertising is an impersonal form of communication that reaches out 
+to the recipients through mass media. Advertising mainly focuses on the product, specific product 
+features, added services, price, packaging unit, trademark, logo, value and ideas worth considering 
+from a social point of view (CSR). Kotler and Keller (2012) are committed to a narrower 
+interpretation of advertising stating that advertising is only related to products, brands and/or 
+services. In advertising, recipients (target group members) are usually aware of the fact that the 
+main intention of marketers with the ads is to persuade and influence their behaviour. Since 
+companies use advertising channels to relay commercials, their target group members can be 
+reached indirectly. In this respect, content marketing is quite different.  According to Kotler et al 
+(2017), content marketing communicates with the marketer’s own public. Content marketing also 
+79%
+70%
+63%
+62%
+58%
+53%
+53%
+49%
+39%
+37%
+0%
+10%
+20%
+30%
+40%
+50%
+60%
+70%
+80%
+90%
+create brand awareness
+educate audiences
+generate demand/leads
+build credibility/trust
+nurture subscribers/audience/leads
+drive attendance to one or more in-person events
+build loyalty with existing clients
+support the launch of a new product
+generate sales/revenue
+build a subscribed audience
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+114 
+ 
+has an appropriately distinguished and defined target audience that receives more personalized 
+content (Hajdú 2018).  
+Kotler et al (2017) express that the concept of traditional media is "one to many", while content 
+marketing, especially social media, almost always mean two-way interactions. Furthermore, 
+advertising helps to sell the product, while content marketing helps the customers to solve their 
+problems and achieve their individual goals. According to Kotler et al (2017), consumers are ready 
+to share the content, while the traditional ads, which are limited in time and space, are rather 
+"skimmed over" by the target audience. It is almost sure to say that advertisements disturb a lot of 
+people since they interrupt their favorite series, or delay videos they want to watch instantly, or 
+fill their mailboxes with emails. Therefore, we can conclude that advertising has an intervening 
+feature. Content marketing aims to maintain a lasting relationship with the target population 
+(Pažėraitė and Repovienė 2018), while advertising is often seasonal and campaign-based (Kotler-
+Keller, 2012). Table 1 illustrates the main differences between advertising and content marketing. 
+So, as Scott (2013) concluded, marketers can buy attention (advertising) or can own attention by 
+creating something interesting and valuable that is published online for free (content marketing). 
+ 
+Table 1.: The comparison of the traditional advertising and content marketing 
+ 
+traditional advertising 
+content marketing 
+type of communication 
+one-way: "I speak only" 
+two-way: "let's talk" 
+purpose of communication  
+promotion of products, 
+brands and services 
+solving the customer's 
+problem at no cost  
+perception of communication from 
+the customer's viewpoint 
+intervening, disturbing 
+giving a helping hand 
+reach 
+a wide range of the 
+population 
+individuals or groups  
+standardization level  
+standardized 
+and 
+impersonal 
+specified 
+and 
+more 
+personalized  
+target groups 
+not own 
+own 
+time span of communication 
+  
+short 
+and 
+campaign-
+based 
+a lasting relationship 
+limitation 
+ 
+limited 
+free 
+target group reaction  
+rejection, 
+skimming 
+over 
+sharing 
+Source: Own compilation based on Kotler et al, 2017; Horváth and Bauer, 2013; Hajdú, 2018; 
+Maczuga et al, 2015; Pažėraitė and Repovienė, 2018 
+ 
+The relationship between direct marketing and content marketing 
+ 
+Direct marketing (DM) is an addressed and interactive form of communication. It aims to achieve 
+measurable responses, which can be orders, purchases, inquiries, or donations. Direct marketing 
+is essentially built on databases. "It allows the potential customers to obtain information, it helps 
+to establish the popularity of a brand or induces immediate purchases" (Horváth and Bauer, 2013, 
+pp. 242.). The fact that direct marketing is built on databases implies that the customer value can 
+be targeted quite accurately. Also, this marketing communication tool is easily optimizable. 
+Telemarketing, mail advertisement, direct mail and direct response advertising are the forms of 
+direct marketing (Horváth and Bauer, 2013). 
+Building brand awareness and credibility are definitely a common point in direct marketing and 
+content marketing. However, direct marketing is less digital than content marketing. In general, 
+the internet as a medium is less dominant in direct marketing, except for e-mail marketing. The 
+purpose of communication in direct marketing is to present the product to make bids. Therefore, 
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+ 
+115 
+ 
+direct marketing is usually related to selling (receiving orders); the eye-catching presentation of 
+products (catalogs) and advertising (mail advertisement). 
+According to Tapp (1999, pp. 23) "direct marketing is rather a sales system than a communication 
+tool". Although nowadays direct marketing has widely been accepted as a marketing 
+communication tool, its sales function cannot be ignored. This point of view is also appeared in 
+Kotler and Keller (2012). According to Horváth and Bauer (2013), direct marketing provides the 
+recipient with a clear opportunity to respond and directly targets the previously defined target 
+groups. Although it also has a pre-defined target group (Hajdú 2018), content marketing places 
+less emphasis on the sales-related responses. In content marketing, the responses affect the content 
+itself. In content marketing, building trust, solving the customer's problem and providing further 
+contents contribute to initiating purchases (Barker 2017). 
+There is another significant difference between direct marketing and content marketing. Direct 
+marketing advertises a product or a service in a targeted manner to increase sales volume through 
+immediate selling. That is why direct marketing is also called "direct order marketing", or “direct 
+advertising". Consequently, direct marketing focuses only on the product, which offers the value 
+for the customer (Kotler and Keller, 2012). Content marketing creates value and provides 
+consumers with it. However, content marketing does not aim to sell immediately, only in one step 
+(Fivetechnology, 2019), it has got longer time-orientation.  
+Combining direct marketing with content marketing can be very effective. If a customer registers 
+an account online, he or she can receive free content (e.g. an ebook), which is content marketing, 
+however, the data provided during the registration are also used to build a database, which can be 
+used for direct marketing purposes.  Content marketing that builds an audience not only identify 
+demands but also generate it. 
+ 
+The relationship between personal selling and content marketing 
+ 
+Few researchers have addressed the question how personal selling and content marketing can be 
+connected. Personal selling is a face-to-face selling technique where the emphasis is on personal 
+interaction. In an event, which can be related to personal selling or could be a content marketing 
+format, the company (brand) and its potential and existing customers can meet in person and/or 
+online. However, it is important to note that the event is only one of several content marketing 
+types, which are mostly digital.  
+Nowadays, the theory of selling as the most important task of the sales staff has already become 
+outdated since the sales department is usually responsible for many other tasks, such as searching 
+for potential customers, providing information, choosing the target market, providing services, 
+collecting information and distribution (Kotler-Keller 2012, pp. 637). 
+Information that the sales staff provide about the products and services, in principle, can refer to 
+the content marketing. Furthermore, services can also link personal selling and content marketing 
+when the sales personnel try to solve the customer's problem.  
+Personal selling and content marketing can sometimes be combined but they can hardly be fell 
+into one category due to the fundamental differences in their characteristics.  
+  
+The relationship between public relations and content marketing 
+ 
+Content marketing should not be confused with public relations (Percy, 2018). In many cases, 
+content marketing is a communication form used on a regular (daily or weekly) bases (Insights 
+2018). Content marketing aims to be part of the consumer's life and seeks to provide value to the 
+customers in an educating and entertaining manner (Lindström and Jörnéus, 2016).  
+Public relation (PR) is a strategic tool aiming to turn brand messages into stories that are appealing 
+to the media and its target audiences (Konczosné Szombathelyi 2018). Thus, PR builds credibility 
+and trust among the stakeholders (Horváth-Bauer 2013). Since public relations is not sales-
+oriented, it is the changes in the mindset of the target audience that should be measured, not its 
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+116 
+ 
+effects on sales (Józsa et al, 2005). PR seeks to build a good reputation of the company; promote 
+the success of the brand; deals with counselling and consulting. All these goals are very similar to 
+those of content marketing, which among other things aims to build credibility and trust. However, 
+content marketing is not a replacement for public relations (Mathewson and Moran, 2016)  
+Józsa et al (2005) emphasize that whatever the goal of PR is, the focus should be on creating trust 
+by emphasizing understanding and willingness to cooperate to gain support from the stakeholders 
+of the company. Trust is also a key factor in building strong brands. Both PR and content 
+marketing can be regarded as regular and systematic communication activities (Józsa et al 2005; 
+Muotsos 2017), and both use rather similar tools such as articles, newsletters, blogs, publications, 
+social media, statistics, e-books, events, etc. (Probusiness, 2018).  
+However, there are some differences between PR and content marketing. Although trust is 
+essential in PR, counselling is only a PR tool or technique. Counseling in PR refers to how we 
+communicate with our clients. It is a recommended course of action that will serve the client’s 
+goals. On the contrary, in content marketing, the valuable content is always provided in the form 
+of education, relevant information or entertainment (Lindström és Jörnéus, 2016). 
+The problem of measuring the effect of PR on sales is also a major difference. The impact of 
+content marketing on sales is a lot easier to measure (Hajdú, 2018), moreover, one of the explicit 
+goals of content marketing is to convert the target public into customers (Barker 2017). 
+Effectiveness of content marketing can easily be measured due to its digital nature. Content 
+marketing is customer-centred, focuses only on selected stakeholders and seeks to solve the 
+customer’s problem by providing information or educational content in an entertaining way 
+(Lindström and Jörnéus 2016; Duc Le 2016). In content marketing, the goal is not to provide all 
+the information but only the relevant content (Wang et al 2017). According to Hajdú (2018), 
+content marketing is a profit-oriented tactical activity to gain customers and make deals. This 
+means that content marketing acquires customers within a reasonable time-period. Content 
+marketing not only produces content, but it also distributes it through its own channels, whereas 
+PR works quite differently in this respect. 
+It is advisable is to combine content marketing with PR since they complement each other. PR 
+can help marketers to make a better story about the brand (Spencer 2014). 
+ 
+The relationship between sales promotion and content marketing 
+ 
+There is a scarcity of literature devoted to the analysing the relationship between sales promotion 
+and content marketing. Horváth and Bauer (2013) refer to sales promotion as a direct influence on 
+consumer behaviour and an impetus to action. With reference to Bauer and Berács (2006), they 
+emphasize that the primary goal of sales promotion is to promote product sales. "Sales promotion 
+is a set of short-term incentive tools which aim to make consumers purchase more products more 
+frequently or buy specific products or services" (Kotler-Keller, 2012, pp. 596). Regarding the 
+consumer's benefit, sales promotion tools can be divided into two categories. Utilitarian and 
+hedonistic tools can be distinguished.  The utilitarian tools provide financial benefits (e.g. price 
+discounts), whereas the hedonistic tools are focusing on entertainment, customer experience and 
+loyalty (Yeshin, 2006). Product samples, gifts, contests and events (trade shows and exhibitions), 
+the tools of sales promotion used to create the customer experience (hedonism), are very much 
+related to content marketing (Józsa, 2014).  
+Product samples make it easier for the customers to try the products. It is an important link between 
+content marketing and sales promotion because content marketing also provides customers with 
+free and useful content when offering a solution to the customer's problem. Thereby, the company 
+can demonstrate its competence and excellence by offering the best solutions to the customer’s 
+problem. Gifts are also commonly used in content marketing in the form of free content.  On the 
+contrary, gifts in sales promotion are not free, they are only given to the customers after the 
+purchase (Horváth and Bauer, 2013). 
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+ 
+117 
+ 
+Events can belong to content marketing and sales promotion; and sometimes even to PR, 
+depending on their goals and their implementation (Józsa et al 2005; Dankó, 2008, Kranz-Pulizzi, 
+2011). The content of the event is the decisive factor. In an event, if marketers present information 
+about how the customer could solve his or her problem, it is highly likely to be content marketing. 
+Contests, games or phone applications can also be content marketing tools (Kranz-Pulizzi, 2011). 
+However, contests and games as sales promotion tools are commonly used to increase sales. In 
+this latter case, purchase is often a pre-requisite of entering the contest. 
+  In sales promotion, the customer experience is directly linked to the purchase, while in content 
+marketing it is not the case. According to Yeshin (2006), in sales promotion, customer loyalty is 
+gained through financial benefits and consumption (e. g. through loyalty points, gifts, etc.). On 
+the contrary, content marketing seeks to achieve the same goal by providing free content that is 
+useful and/or entertaining. The primary goal of sales promotion is to increase product sales, which 
+can be a distinguishing factor between content marketing and sales promotion. Although content 
+marketing is also sales-oriented in the long run, here the deal is achieved in several steps 
+(Fivetechnology, 2019). In this process, the very first step is building trust by giving value without 
+asking for compensation or purchase (Repoviener, 2017; Maczuga et al, 2015). We can conclude 
+that the main differences between content marketing and sales promotion can be found in their 
+objectives and time-orientation. Content marketing, which is not a short-term tool, is often 
+regarded as is an introductory stage of sales as it does not aim to make purchases quickly. 
+ 
+Conclusions 
+ 
+This paper investigates the relationship between content marketing and the five traditional 
+marketing communication tools.  The goal of the article is to generate a discussion on the status 
+of content marketing. In this paper, content marketing was compared to advertising, direct 
+marketing, personal selling, public relations and sales promotion to find out the main differences 
+and similarities. An extensive literature review explored some fundamental differences between 
+the traditional marketing communication tools and content marketing. Based on this result, content 
+marketing can be regarded as a novel marketing communication tool and the sixth element of the 
+revised marketing communication mix. Content marketing can be effectively used in marketing 
+campaigns in the digital environment. Because of its digital nature, content marketing can be more 
+effective in digitally advanced target markets. One of the positive effects of COVID-19 is the 
+accelerated digitalization, which favorable to the use of content marketing.  
+ 
+Acknowledgements 
+ 
+“The described article/presentation/study was carried out as part of the EFOP-3.6.1-16-2016-
+00011 “Younger and Renewing University – Innovative Knowledge City – institutional 
+development of the University of Miskolc aiming at intelligent specialisation” project 
+implemented in the framework of the Szechenyi 2020 program. The realization of this project is 
+supported by the European Union, co-financed by the European Social Fund.” 
+"A cikkben/előadásban/tanulmányban ismertetett kutató munka az EFOP-3.6.1-16-2016-00011 
+jelű „Fiatalodó és Megújuló Egyetem – Innovatív Tudásváros – a Miskolci Egyetem intelligens 
+szakosodást szolgáló intézményi fejlesztése” projekt részeként – a Széchenyi 2020 keretében – az 
+Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg" 
+ 
+ 
+ 
+ 
+
+Észak-magyarországi Stratégiai Füzetek XVIII. évf.  2021  1 
+118 
+ 
+References  
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+

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+page_content='Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1 110  Szabolcs Nagy – Gergő Hajdú  The relationship between content marketing and the traditional marketing communication tools  Digitalization is making a significant impact on marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' New marketing approaches and tools are emerging which are not always clearly categorised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' This article seeks to investigate the relationship between one of the novel marketing tools, content marketing, and the five elements of the traditional marketing communication mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Based on an extensive literature review, this paper analyses the main differences and similarities between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' This article aims to generate a debate on the status of content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" According to the authors' opinion, content marketing can be considered as the sixth marketing communication mix element." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, further research is needed to fill in the existing knowledge gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Keywords: content marketing, trends, advertising, sales promotion, direct marketing, personal selling, public relations JEL: M31, M37  https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='32976/stratfuz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='25  Introduction  Digitalization and the ongoing information technology revolution pose remarkable possibilities and challenges for marketing (Piskóti, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Due to digitalization, consumer behaviour is constantly changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Consumers’ stimulus threshold is increasing because of the greater exposure to information (Kotler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' At the same time, smart devices are becoming increasingly dominant (Nagy, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' E-commerce (Nagy, 2016), and the various social networks are becoming popular (Sethi, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' These trends accelerate the emergence of new methods and trends in marketing (Nagy, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It is advisable for marketers to understand how those new methods and tools work since they help to reach out to consumers to influence their behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, the lack of advanced information technology in Hungary poses some problems in this process (Kamaraonline, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Marketing communication tools can often be divided into two main groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Traditional and digital solutions can be distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, according to Kotler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2017), the two categories have recently been merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing is essentially a digital solution having some offline features as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The significance of content marketing is supported by Kotler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2017), who found - referring to the research findings of Content Marketing Institute and the MarketingProfs - that 76% of the B2C companies and 88% of the B2B companies used content marketing in North America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Furthermore, B2C companies spent 32% of their marketing budgets on content marketing, while B2B companies spent 28%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 57% of the B2C companies increased their content marketing budget by at least 1%, while 29% of them did not change the budget (Brenner 2019, based on Content Marketing Institute 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The companies mainly increased their content marketing budgets in the following areas: content production (56%), content marketing personnel (37%), paid distribution of content (36%), content marketing technology (29%), and content marketing outsourcing (29%) (Murton Beets 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' These facts also underline the importance of content marketing in today’s digital world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' If we accept that traditional and digital solutions have been merging (Kotler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', 2017), it means that the traditional classification of marketing communication tools should be revised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The communication tools should rather be classified according to their functions and operating mechanisms than according to the type of technological solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' From this perspective, content marketing (CM) is a new approach to marketing communication and a novel marketing communication tool that can be combined with traditional marketing tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Therefore, the present paper seeks to investigate the relationship between content marketing and the five, traditional  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1  111  marketing communication tools to generate discussion if content marketing is the sixth element of the revised marketing communication mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Literature review  Content marketing definition, functions, and spending  Content marketing is the creation and distribution of relevant, timely, and valid content (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Its primary purpose is to create customer trust and value (Repoviener, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing may have entertaining or educational functions (Duc Le 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Lindström and Jörnéus, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing can be effectively used both in B2C and B2B markets (Iankova et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Kotler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2017), the content can serve brand-building or sales promotion purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" According to Moutsos (2019), 55% of the companies were capable of generating sales and income, and 53% of them were capable of increasing their existing customers' loyalty through content marketing in 2018." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" So, content marketing can be used to generate income and sales, and also, to increase customers' loyalty." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Content types and formats  Content marketing may appear in various formats based on the type of content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It could be audio and/or visual content (videos, live streaming, webinars);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' written digital content (articles, blogs, ebooks), images (infographics, photos, GIFs, charts), in-person content (events, presentations, workshops);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' audio-only digital content (podcasts, audiobooks), and written print content (magazines, books, brochures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' shows the different types of content and how B2B marketers changed their use of content types/formats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' shows the very same trends in B2C markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The change of use of content types/format in B2B markets Source: Own compilation based on Murton Beets, 2018  As Figure 1 illustrates, in B2B markets, the use of audio/visual content;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' written digital content and images became more popular, while the use of written print content significantly decreased compared to the other types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The same trends can be seen in the B2C markets (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The only slight difference between the two markets is in the use of audio-only digital content, which significantly dropped in the B2C market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  64%  61%  56%  41%  38%  27%  32%  33%  40%  52%  54%  56%  4%  6%  4%  7%  8%  17%  Audio/Visual Content  Written Digital Content  Images  In Person Content  Audio only Digital Content  Written Print Content  Increased  Remained the same  Decreased  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1 112  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The change of use of content types/format in B2C markets Source: Own compilation based on Murton Beets, 2018  In practice, various types of content can be used to reach out to consumers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' As far as the type of content concerned, e-mail campaign is the most popular one, used by 87% of the companies (Murton Beets 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, the following content types are also frequently used (values in brackets show the percentage of companies using the given content type): educative content (77%), actions calling for the next step (62%), events involving personal interactions (61%), telling stories (45%), offers (27%) and community building involving the public (23%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Trends and forecasts are less popular, only 5% of the companies used them (Murton Beets 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  The goals of content marketing  Content marketing helps to achieve several goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The goal of content marketing is to gain customers (Barker, 2017) and to build customer relationships (Pažėraitė and Repovienė, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing can very effectively be used to create brand awareness, educate audiences, generate demand/leads, and build credibility/trust (Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Also, content marketing is an effective tool for nurturing subscribers/audience/leads;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' driving attendance to one or more in-person events, building loyalty with existing clients, and supporting the launch of a new product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It can even be used to achieve sales/revenue generation and build a subscribed audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' presents the possible goals companies managed to successfully achieve by using content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  69%  64%  63%  37%  30%  27%  25%  31%  30%  51%  48%  63%  6%  5%  7%  12%  22%  10%  0%  10%  20%  30%  40%  50%  60%  70%  80%  90%  100%  Audio/Visual Content  Written Digital Content  Images  In Person Content  Audio only Digital Content  Written Print Content  Increased  Remained the same  Decreased  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1  113  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing goals Source: Own compilation based on Content Marketing Institute (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Note: Goals enterprise marketers have achieved by using content marketing successfully  Research methodology  This paper seeks to generate a debate on the current state of content marketing, and it aims to create a base for future quantitative research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It synthesizes the relevant literature to analyze the relationship between content marketing and the traditional marketing communication tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It makes an attempt to distinguish content marketing from the other elements in marketing communication mix, which are advertising, sales promotion (SP), public relations (PR), personal selling, and direct marketing (DM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In the following section, based on extensive literature review, the five traditional marketing communication tools are compared to content marketing to reveal the similarities and differences between them regarding the type, purpose, standardization, time span and reach of communication and the target groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Research findings and discussion  The relationship between advertising and content marketing  Advertising is the most prominent element of the traditional communication mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Horváth and Bauer (2013) advertising is an impersonal form of communication that reaches out to the recipients through mass media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Advertising mainly focuses on the product, specific product features, added services, price, packaging unit, trademark, logo, value and ideas worth considering from a social point of view (CSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Kotler and Keller (2012) are committed to a narrower interpretation of advertising stating that advertising is only related to products, brands and/or services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In advertising, recipients (target group members) are usually aware of the fact that the main intention of marketers with the ads is to persuade and influence their behaviour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Since companies use advertising channels to relay commercials, their target group members can be reached indirectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In this respect, content marketing is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Kotler et al (2017), content marketing communicates with the marketer’s own public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing also 79% 70% 63% 62% 58% 53% 53% 49% 39% 37% 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% create brand awareness educate audiences generate demand/leads build credibility/trust nurture subscribers/audience/leads drive attendance to one or more in-person events build loyalty with existing clients support the launch of a new product generate sales/revenue build a subscribed audience  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1 114  has an appropriately distinguished and defined target audience that receives more personalized content (Hajdú 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Kotler et al (2017) express that the concept of traditional media is "one to many", while content marketing, especially social media, almost always mean two-way interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Furthermore, advertising helps to sell the product, while content marketing helps the customers to solve their problems and achieve their individual goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Kotler et al (2017), consumers are ready to share the content, while the traditional ads, which are limited in time and space, are rather "skimmed over" by the target audience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It is almost sure to say that advertisements disturb a lot of people since they interrupt their favorite series, or delay videos they want to watch instantly, or fill their mailboxes with emails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Therefore, we can conclude that advertising has an intervening feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing aims to maintain a lasting relationship with the target population (Pažėraitė and Repovienė 2018), while advertising is often seasonal and campaign-based (Kotler- Keller, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Table 1 illustrates the main differences between advertising and content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' So, as Scott (2013) concluded, marketers can buy attention (advertising) or can own attention by creating something interesting and valuable that is published online for free (content marketing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=': The comparison of the traditional advertising and content marketing  traditional advertising content marketing type of communication one-way: "I speak only" two-way: "let\'s talk" purpose of communication promotion of products,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" brands and services solving the customer's problem at no cost perception of communication from the customer's viewpoint intervening," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' disturbing giving a helping hand reach a wide range of the population individuals or groups standardization level standardized and impersonal specified and more personalized target groups not own own time span of communication  short  and  campaign based  a lasting relationship  limitation  limited free target group reaction rejection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' skimming over sharing Source: Own compilation based on Kotler et al,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Horváth and Bauer, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Hajdú, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Maczuga et al, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Pažėraitė and Repovienė, 2018  The relationship between direct marketing and content marketing  Direct marketing (DM) is an addressed and interactive form of communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It aims to achieve measurable responses, which can be orders, purchases, inquiries, or donations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Direct marketing is essentially built on databases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' "It allows the potential customers to obtain information, it helps to establish the popularity of a brand or induces immediate purchases" (Horváth and Bauer, 2013, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The fact that direct marketing is built on databases implies that the customer value can be targeted quite accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Also, this marketing communication tool is easily optimizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Telemarketing, mail advertisement, direct mail and direct response advertising are the forms of direct marketing (Horváth and Bauer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Building brand awareness and credibility are definitely a common point in direct marketing and content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, direct marketing is less digital than content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In general, the internet as a medium is less dominant in direct marketing, except for e-mail marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The purpose of communication in direct marketing is to present the product to make bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Therefore,  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1  115  direct marketing is usually related to selling (receiving orders);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' the eye-catching presentation of products (catalogs) and advertising (mail advertisement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Tapp (1999, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 23) "direct marketing is rather a sales system than a communication tool".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Although nowadays direct marketing has widely been accepted as a marketing communication tool, its sales function cannot be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' This point of view is also appeared in Kotler and Keller (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Horváth and Bauer (2013), direct marketing provides the recipient with a clear opportunity to respond and directly targets the previously defined target groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Although it also has a pre-defined target group (Hajdú 2018), content marketing places less emphasis on the sales-related responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In content marketing, the responses affect the content itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" In content marketing, building trust, solving the customer's problem and providing further contents contribute to initiating purchases (Barker 2017)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' There is another significant difference between direct marketing and content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Direct marketing advertises a product or a service in a targeted manner to increase sales volume through immediate selling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' That is why direct marketing is also called "direct order marketing", or “direct advertising".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Consequently, direct marketing focuses only on the product, which offers the value for the customer (Kotler and Keller, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing creates value and provides consumers with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  However, content marketing does not aim to sell immediately, only in one step (Fivetechnology, 2019), it has got longer time-orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Combining direct marketing with content marketing can be very effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' If a customer registers an account online, he or she can receive free content (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' an ebook), which is content marketing, however, the data provided during the registration are also used to build a database, which can be used for direct marketing purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing that builds an audience not only identify demands but also generate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  The relationship between personal selling and content marketing  Few researchers have addressed the question how personal selling and content marketing can be connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Personal selling is a face-to-face selling technique where the emphasis is on personal interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In an event, which can be related to personal selling or could be a content marketing format, the company (brand) and its potential and existing customers can meet in person and/or online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, it is important to note that the event is only one of several content marketing types, which are mostly digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Nowadays, the theory of selling as the most important task of the sales staff has already become outdated since the sales department is usually responsible for many other tasks, such as searching for potential customers, providing information, choosing the target market, providing services, collecting information and distribution (Kotler-Keller 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 637).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Information that the sales staff provide about the products and services, in principle, can refer to the content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" Furthermore, services can also link personal selling and content marketing when the sales personnel try to solve the customer's problem." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Personal selling and content marketing can sometimes be combined but they can hardly be fell into one category due to the fundamental differences in their characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  The relationship between public relations and content marketing  Content marketing should not be confused with public relations (Percy, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In many cases, content marketing is a communication form used on a regular (daily or weekly) bases (Insights 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" Content marketing aims to be part of the consumer's life and seeks to provide value to the customers in an educating and entertaining manner (Lindström and Jörnéus, 2016)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Public relation (PR) is a strategic tool aiming to turn brand messages into stories that are appealing to the media and its target audiences (Konczosné Szombathelyi 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Thus, PR builds credibility and trust among the stakeholders (Horváth-Bauer 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Since public relations is not sales- oriented, it is the changes in the mindset of the target audience that should be measured, not its  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1 116  effects on sales (Józsa et al, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' PR seeks to build a good reputation of the company;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' promote the success of the brand;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' deals with counselling and consulting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' All these goals are very similar to those of content marketing, which among other things aims to build credibility and trust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, content marketing is not a replacement for public relations (Mathewson and Moran, 2016) Józsa et al (2005) emphasize that whatever the goal of PR is, the focus should be on creating trust by emphasizing understanding and willingness to cooperate to gain support from the stakeholders of the company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Trust is also a key factor in building strong brands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Both PR and content marketing can be regarded as regular and systematic communication activities (Józsa et al 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Muotsos 2017), and both use rather similar tools such as articles, newsletters, blogs, publications, social media, statistics, e-books, events, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (Probusiness, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, there are some differences between PR and content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Although trust is essential in PR, counselling is only a PR tool or technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Counseling in PR refers to how we communicate with our clients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It is a recommended course of action that will serve the client’s goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' On the contrary, in content marketing, the valuable content is always provided in the form of education, relevant information or entertainment (Lindström és Jörnéus, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The problem of measuring the effect of PR on sales is also a major difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  The impact of content marketing on sales is a lot easier to measure (Hajdú, 2018), moreover, one of the explicit goals of content marketing is to convert the target public into customers (Barker 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Effectiveness of content marketing can easily be measured due to its digital nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing is customer-centred, focuses only on selected stakeholders and seeks to solve the customer’s problem by providing information or educational content in an entertaining way (Lindström and Jörnéus 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Duc Le 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In content marketing, the goal is not to provide all the information but only the relevant content (Wang et al 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Hajdú (2018), content marketing is a profit-oriented tactical activity to gain customers and make deals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' This means that content marketing acquires customers within a reasonable time-period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing not only produces content, but it also distributes it through its own channels, whereas PR works quite differently in this respect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' It is advisable is to combine content marketing with PR since they complement each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' PR can help marketers to make a better story about the brand (Spencer 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  The relationship between sales promotion and content marketing  There is a scarcity of literature devoted to the analysing the relationship between sales promotion and content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Horváth and Bauer (2013) refer to sales promotion as a direct influence on consumer behaviour and an impetus to action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' With reference to Bauer and Berács (2006), they emphasize that the primary goal of sales promotion is to promote product sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' "Sales promotion is a set of short-term incentive tools which aim to make consumers purchase more products more frequently or buy specific products or services" (Kotler-Keller, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 596).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" Regarding the consumer's benefit, sales promotion tools can be divided into two categories." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Utilitarian and hedonistic tools can be distinguished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The utilitarian tools provide financial benefits (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' price discounts), whereas the hedonistic tools are focusing on entertainment, customer experience and loyalty (Yeshin, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Product samples, gifts, contests and events (trade shows and exhibitions), the tools of sales promotion used to create the customer experience (hedonism), are very much related to content marketing (Józsa, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Product samples make it easier for the customers to try the products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" It is an important link between content marketing and sales promotion because content marketing also provides customers with free and useful content when offering a solution to the customer's problem." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Thereby, the company can demonstrate its competence and excellence by offering the best solutions to the customer’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Gifts are also commonly used in content marketing in the form of free content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' On the contrary, gifts in sales promotion are not free, they are only given to the customers after the purchase (Horváth and Bauer, 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1  117  Events can belong to content marketing and sales promotion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' and sometimes even to PR, depending on their goals and their implementation (Józsa et al 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Dankó, 2008, Kranz-Pulizzi, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The content of the event is the decisive factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In an event, if marketers present information about how the customer could solve his or her problem, it is highly likely to be content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Contests, games or phone applications can also be content marketing tools (Kranz-Pulizzi, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' However, contests and games as sales promotion tools are commonly used to increase sales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In this latter case, purchase is often a pre-requisite of entering the contest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In sales promotion, the customer experience is directly linked to the purchase, while in content marketing it is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' According to Yeshin (2006), in sales promotion, customer loyalty is gained through financial benefits and consumption (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' through loyalty points, gifts, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' On the contrary, content marketing seeks to achieve the same goal by providing free content that is useful and/or entertaining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The primary goal of sales promotion is to increase product sales, which can be a distinguishing factor between content marketing and sales promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Although content marketing is also sales-oriented in the long run, here the deal is achieved in several steps (Fivetechnology, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  In this process, the very first step is building trust by giving value without asking for compensation or purchase (Repoviener, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Maczuga et al, 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' We can conclude that the main differences between content marketing and sales promotion can be found in their objectives and time-orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing, which is not a short-term tool, is often regarded as is an introductory stage of sales as it does not aim to make purchases quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Conclusions  This paper investigates the relationship between content marketing and the five traditional marketing communication tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The goal of the article is to generate a discussion on the status of content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' In this paper, content marketing was compared to advertising, direct marketing, personal selling, public relations and sales promotion to find out the main differences and similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' An extensive literature review explored some fundamental differences between the traditional marketing communication tools and content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Based on this result, content marketing can be regarded as a novel marketing communication tool and the sixth element of the revised marketing communication mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content marketing can be effectively used in marketing campaigns in the digital environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Because of its digital nature, content marketing can be more effective in digitally advanced target markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' One of the positive effects of COVID-19 is the accelerated digitalization, which favorable to the use of content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Acknowledgements  “The described article/presentation/study was carried out as part of the EFOP-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='1-16-2016- 00011 “Younger and Renewing University – Innovative Knowledge City – institutional development of the University of Miskolc aiming at intelligent specialisation” project implemented in the framework of the Szechenyi 2020 program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' The realization of this project is supported by the European Union, co-financed by the European Social Fund.” "A cikkben/előadásban/tanulmányban ismertetett kutató munka az EFOP-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='1-16-2016-00011 jelű „Fiatalodó és Megújuló Egyetem – Innovatív Tudásváros – a Miskolci Egyetem intelligens szakosodást szolgáló intézményi fejlesztése” projekt részeként – a Széchenyi 2020 keretében – az Európai Unió támogatásával, az Európai Szociális Alap társfinanszírozásával valósul meg"  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1 118  References  BARKER, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' How to Create High-Converting Content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Retrieved from https://contentmarketinginstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='com/2017/05/create-high-converting-content, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' BRENNER, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Content Marketing Survey: Marketers Focus On Content Creation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Retrieved from: https://marketinginsidergroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='com/content-marketing/2019-content- marketing-survey-content-creation/, 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' CONTENT MARKETING INSTITUTE (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Enterprise Content Marketing 2019 – Benchmarks, Budgets, and Trends – North America.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Retrieved from https://contentmarketinginstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='com/wp-content/uploads/2019/02/FINAL- 2019_Enterprise_Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='pdf,  05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' DANKÓ, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Értékesítés-ösztönzés.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' Pro Marketing Miskolc Egyesület DUC LE, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' Retrieved from: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content='  2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' HAJDÚ, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Az online marketingcontrolling ��rtékelési folyamata a tartalommarketing ROI segítségével.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' 5-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='24387/CI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2 HORVÁTH, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - BAUER, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Marketingkommunikáció – Stratégia, új média, fogyasztói részvétel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Budapest: Akadémiai Kiadó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' IANKOVA S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - DAVIES I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - ARCHER-BROWN C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - MARDER B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - YAU A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' A comparison of social media marketing between B2B, B2C and mixed business models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Industrial marketing management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', 81, 169-179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='001 JÓZSA, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', PISKÓTI, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', REKETTYE, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', VERES, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Döntésorientált marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' KJK KERSZÖV Jogi és Üzleti kiadó Kft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Budapest KAMARAONLINE (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Versenyhátrányt okoz a magyar vállalkozásoknak az informatikai lemaradás.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content='hu/cikk/versenyhatranyt-okoz-a- magyar-vallalkozasoknak-az-informatikai-lemaradas,  30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' KONCZOSNÉ SZOMBATHELYI, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' A PR úttörői és napjainkig tartó hatásuk, Széchenyi István Egyetem, Győr, Retrieved from: https://kgk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='sze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content='pdf,   05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' -  KELLER, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Marketingmenedzsment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Budapest: Akadémiai Kiadó.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' KOTLER, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - KARTAJAYA H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - SETIAWAN I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Marketing 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='0 - Moving from Traditional to Digital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' John, & Sons, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' KRANZ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=', PULIZZI J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' Content Marketing Playbook, Junta42 Conent Marketing Institute & Kranz Communications LINDSTRÖM, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - JÖRNÉUS, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Co-Creating value through Content Marketing, University of Gothengurg, School of Business, Economics and Law MACZUGA P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - SIKORSKA K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' Content marketing Handbook: Simple Ways to Innovate Your Marketing Approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Project under Lifelong Learning Programme of European Commission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Warsaw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Racom Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' ISBN: 978-83-63481-10-0 MATHEWSON, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - MORAN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Outside-in marketing: Using big data to guide your content marketing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Boston: IBM Press, Pearson plc MOUTSOS, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=" Publishing frequency: why (and how) we're changing things up." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Retrieved from: https://contentmarketinginstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='com/2017/12/publishing-frequency-changing/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' MOUTSOS, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Tech Content Marketers Talk Content Creation Challenges, Tools, and Trends Retrieved from: https://contentmarketinginstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='com/2019/03/tech-content- marketers-research/,  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='  Észak-magyarországi Stratégiai Füzetek XVIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' évf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' \uf0e0 2021 \uf0e0 1  119  MURTON BEETS, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' 2019 B2B Content Marketing Research: It Pays to Put Audience First.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' Retrieved from: https://contentmarketinginstitute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content='com/2018/10/research-b2b- audience/,  11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' NAGY, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+page_content=' WANG, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
+page_content=' - MALTHOUSE, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfVPwp/content/2301.01279v1.pdf'}
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+URSI GASS 2023, Sapporo, Japan, 19 – 26 August 2023
+Deep Learning Model with Attention Mechanism for Super-resolution of Wireless Channel
+Characteristics
+Haoyang Zhang(1), Xiping Wang (1), and Danping He(2)
+(1) ABC University, Prague, Czechia; e-mail: [email protected]; [email protected]
+(2) The Next University, Neverland, USA; e-mail: [email protected]
+Abstract
+As an emerging approach, deep learning plays an increas-
+ingly influential role in channel modeling. In this paper, we
+present a super-resolution (SR) model for channel charac-
+teristics. Residual connection and attention mechanism are
+applied to this convolutional neural network (CNN) model.
+Experiments prove that the proposed model can reduce the
+noise interference generated in the super-resolution process
+of channel characteristics and reconstruct low-resolution
+data with high accuracy. The mean absolute error of our
+channel SR model on the PL achieves the effect of 2.82 db
+with scale factor 2. Compared with traditional ray tracing
+methods and vision transformer (ViT), the proposed model
+demonstrates less running time and computing cost.
+1
+Introduction
+With the rapid development of wireless communication
+technology, 5G is widely used in various fields and daily
+life applications, such as media streaming, gaming, and
+video conferencing[1]. Accurate channel model is regarded
+as the foundation of future wireless network. Generally,
+deterministic and semi-deterministic modeling are the two
+mainstream methods of channel modeling[2].
+However,
+this method has many limitations, including low computa-
+tional efficiency, excessive computing power consumption,
+and it requires complicated simulation.
+Many researchers are trying to make breakthroughs in chan-
+nel modeling by machine learning (ML)[3]. Because of its
+generalizable architecture, machine learning is widely used
+in almost every branch of science and technology. Chan-
+nel modeling is not an exception[4]. As for ML methods,
+deep learning (DL) models of super-resolution (SR) such as
+CNN [5], Transformer [6] and generative adversarial net-
+work (GAN) [7] are frequently employed. However, the
+scope of current research is still limited. Most studies are
+based on pure electromagnetic environment data without
+considering complex terrain and building distribution.
+In this paper, we propose a DL model of SR for channel
+characteristics.
+It recovers high-resolution (HR) charac-
+teristics over low-resolution (LR) scenes built from origi-
+nal data. We transform the channel characteristics gener-
+ation problem into a super-resolution problem of feature
+maps containing various electromagnetic parameters. This
+problem is addressed by using CNN with residual connec-
+tion and attention mechanisms. The overview of our pro-
+posed model is shown in Fig. 1. We used CloudRT[8] for
+ray tracing simulation, simulating the propagation of radio
+waves in complex environments in dense urban areas, and
+obtained channel characteristic information. Besides mean
+absolute error (MAE), we also incorporate the standard de-
+viation error (STDE) into the loss function to balance reli-
+ability and error. We evaluate our proposed model by abla-
+tion study and comparisons with other SR models.
+2
+Methodology
+2.1
+Data preprocessing and construction
+Utilizing CloudRT, we developed a high-precision channel
+feature dataset, RT-urban, which is used for training our
+proposed SR model. Simulation configuration details are
+summarized in[8]. We obtained seven channel characteris-
+tics by simulation which are the height of buildings (h), path
+loss (PL), multipath power ratio (Rp), and LOS/NLOS area
+classification, root mean square (RMS) delay (DS), RMS
+azimuth angle spread (φ), and RMS elevation angle spread
+(θ).The latter six characteristics are our SR targets. 462 ray
+data were generated in more than 70 dense urban areas.
+Values of channel characteristics that are far beyond ordi-
+nary thresholds in communication systems are set as the
+minimum (PL, Rp) or the maximum (DS) of the corre-
+sponding normal range. NaN value represents the chan-
+nel characteristics data of receivers located inside buildings.
+This kind of value should be void but set as a real number
+out of the normal range so that the DL model can distin-
+guish. The related data and processing methods are shown
+in our previous work[8].
+2.2
+Neural network architecture
+The super-resolution problem of the channel feature can be
+expressed as a super-resolution problem of the image. The
+key to image super-resolution lies in recovering from LR
+data to HR data.[9]
+This paper’s copyright is held by Haoyang Zhang. It is published in these proceedings and included in any archive such as
+IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of
+Scholarly Work.”
+arXiv:2301.04479v1  [eess.SP]  11 Jan 2023
+
+Figure 1. The overview of proposed SR model for wireless
+channel characteristics.
+2.2.1
+Deep-shallow pipe based on residual network
+In this study, we want to directionally generate various HR
+channel features from the collected and processed LR chan-
+nel feature data. There are differences between different
+channel characters, and if all characters are fed into our
+network, the results will be unsatisfactory.
+We propose
+a method of classifying the propagation of deep-shallow
+channels.
+In Fig. 2, the deep-shallow backbone has a deep and shal-
+low panel, which extracts features of different dimensions
+from input data. The deep panel has two convolutional lay-
+ers with activation function ReLU more than the shallow
+panel and repeats the illustrated convolution block multiple
+times to extract features fully. The number of this block N
+in our work is 2. We also introduce the method of residual
+connection. It helps to expand the field of view and reduce
+the loss of features caused by excessive convolution opera-
+tions in the process of convolution.
+�
+Hdeep = Fdeep(Iin,Wd)+Iin
+Hshallow = Fshallow(Iin,Ws)+Iin
+(1)
+Where H is the output after convolution, Iin and W present
+the input data and weights learn from deep and shallow pipe
+F. After each batch of convolution block, the unprocessed
+input data is residually connected to the feature maps pro-
+duced by the operation. It can help prevent gradient dis-
+sipation during back-propagation, thus making it easier to
+train deep networks.
+2.2.2
+Attention mechanisms and feature blocks
+It has to be considered that multi-path channel character-
+istics such as PL, DL, and LOS. The learning patterns of
+these features are completely different and tend to show dis-
+tinguished differences in the learning process. So we intro-
+duced an attention mechanism for multi-feature extraction
+on the framework of the SR model in Fig.3. We split the
+feature map F into several branches Klist by using kernels
+with different sizes:
+Klist = Conv(F,kernellist = [1,3,5,7]size)
+(2)
+Figure 2. The overview of the proposed SR model with an
+attention mechanism. Conv(x) means that the convolution
+scale is x. PL* stands for PL, DS, and Rp.
+Figure 3. The overview of attention layer
+As stated before, our goal is to control the information flows
+from multiple characteristics carrying different scales of in-
+formation into the next layer. To achieve this, we need to
+integrate channels from the first N branches of the K list:
+S = ∑
+i
+Ki,
+i = [1,3,5,7]
+(3)
+then we embed the global information by simply using av-
+erage pooling to generate channels as W ∈ RC. The C chan-
+nels are calculated by the following in dimensions h×w:
+Wc = Favg(S) =
+1
+h×w
+h
+∑
+i
+w
+∑
+j
+S(i, j)
+(4)
+So we can get a compact weight w ∈ R64×64 of different
+channels by a simple fully connected layer, with the re-
+duction of dimensionality for better efficiency. By fusing
+branches of features, this mechanism enhances the ability
+of the previous deep and shallow panels to extract char-
+acteristics and it will also pay attention to multi-scales of
+information when dell with different learning patterns of
+characteristics.
+2.2.3
+Loss functions
+We generally use pixel loss instead of content loss for char-
+acteristics learning tasks. Through experiments, we found
+that the L1loss can help achieve better results on our task
+than the peak signal-to-noise ratio.
+In data preparation,
+we mentioned six different channel features in this task.
+However, the characteristic of LOS is unique. It only has
+This paper’s copyright is held by Haoyang Zhang. It is published in these proceedings and included in any archive such as
+IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of
+Scholarly Work.”
+
+DataSet
+Loss function
+high
+Ray-Tracing
+resolution(HR)
+Validation
+LosSi1(HR, lout)
+Down
+sampling
+low
+Input
+SR Model
+iteration
+resolution(LR)
+compare
+evaluate
+Deep -
+Input
+panel
+Channel
+multiply
+attention
+LR Data
+blocks
+Shallow.
+Base Model
+panelH
+Nlos
+Block
+1
+128
+ATT
+64
+64
+Deep feature panel
+Theta
+ayer
+Block
+..
+(c, 200, 200)
+Input
+PL*
+Block
+Shallow feature panel
+F(64, 200, 200)
+Output
+Fine-tune Part
+Transfer
+Reshape
+Conv(x)
+Relu
+Block
+Attention
+Conv(7)
+Conv(64)=wxK
+m
+Kernel 5x5
+M
++
+*N blocks
+F(64. 200. 200)
+S(64, 200, 200)
+*N blocks
+F'(64, 200, 200)
+N3
+W(64, 1, 1
+Kernel 7x7
+Learn w E R64*64
+from full connection
+K(64, 200, 200)
+K'(64, 200, 200)two types of integer values. Therefore, the performance
+can only be judged by valuing the accuracy of the model
+for these two types of numerical classification. As a result
+cross-entropy is used as the evaluation index.
+lossl1(ˆI,I) =
+n
+(hw)2 ∑
+i, j
+| ˆIi, j −Ii, j|
+(5)
+lossce(ˆI,I) = −
+n
+(hw)2 ∑
+i, j
+Ii, j,klogˆIi, j,k
+(6)
+Where n,h,w represents the number of batches to be esti-
+mated and the length and width of the region we chose for
+input data, respectively. In order to enhance the confidence
+of the fitted data, we also include the STDE as part of the
+evaluation index of the fitting effect of the loss function:
+STDE(��I,I) =
+�
+1
+hw ∑
+i, j
+(ˆIi, j −Ii, j)2
+(7)
+The standard deviation reflects the degree of dispersion be-
+tween the pixel value of the image and the mean value. The
+smaller the standard deviation, the higher the fitting degree
+of the image.
+3
+Experiment
+3.1
+Performance of the proposed model
+(a) MAE of Path loss
+(b)
+Classification
+accuracy
+of
+LOS/NLOS
+Figure 4. MAE and classification accuracy of 2 main SR
+targets during the training process.
+It can be observed from the experiment results that, the
+MAE of our channel SR model on the PL achieves the effect
+of 2.82 db with scale factor 2. Under the same conditions,
+the comparison results of ResNet50, Vit, GAN, and UNet
+are shown in Fig. 5 and Table 1. Obviously, after testing,
+it can be found that these DL models, which achieved good
+results in other fields such as computer vision or natural
+language processing, performed worse than the proposed
+model in this paper. On the single-task fitting for PL, the
+performance of ResNet50 and UNet is around 7-8 db on
+average after several epochs of training. Moreover, ViT can
+only reach 8 db after modifying more layers and processing
+with masks. The best result of GAN is even larger than 12
+db and still contains much noise. Our model has achieved
+results far exceeding the popular SR models on the channel
+super-resolution task through the comparison.
+On the RT-Urban dataset we constructed, we performed SR
+training on 6 main channel features with scales of 2, 4, and
+8, and the results are shown in Figure. 4 and Table 2. Dur-
+ing the training process of the shared parameter layer, it
+can be found that the loss of each feature decreases rapidly
+around the first 100 epochs and achieves a relatively sta-
+ble result. The performance of the following targeted fea-
+ture extractor can be optimized by 0.5-0.7 db on the best
+achieved by the backbone after fine-tuning.
+In the classification training of LOS/NLOS and θ, it can
+be observed that the second half of SR training at scale 8
+is more volatile and not as smooth as other features. After
+splitting the feature map data and visual analysis, we found
+that large-scale downsampling will make the edge of the
+classification area seriously jagged. Moreover, extracting a
+more accurate mapping relationship is impossible and will
+affect accuracy. However, the classification can still achieve
+a correct rate of more than 91%, indicating that our model
+has an imposing recovery effect on the channel feature data.
+(a) Ground-Truth
+(b) UNet
+(c) ViT
+(d) Ours
+(e) GAN
+(f) ResNet50
+Figure 5. Visualization of Super-Resolution results of DS
+in a region of 1000 × 1000 m2 with scale factor 2.
+Table 1. Quantitative results for the experiments
+Dataset
+Method
+MAE/db
+STDE
+RT-Urban
+ViT
+7-8
+11-12
+GAN
+12-14
+20-22
+GANSR
+7-8
+17-20
+ResNet50
+6-7
+9-10
+UNet
+6-7
+12-14
+Ours
+2.83 (best)
+5.09
+3.2
+Ablation study
+The ablation experiments are used to verify whether these
+methods we take to improve the SR of the model improve
+This paper’s copyright is held by Haoyang Zhang. It is published in these proceedings and included in any archive such as
+IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of
+Scholarly Work.”
+
+8
+Scale = 2
+Scale = 4
+7
+Scale = 8
+Path Loss (dB)
+3
+50
+100
+150
+200
+0
+Epochs100
+98
+96
+94
+Accuracy (
+92
+90
+88
+86
+Scale = 2
+Scale = 4
+84
+Scale = 8
+82
+100
+150
+200
+50
+0
+EpochsTable 2. Super resolution performance (MAE & STDE)
+Scale
+Targets
+PL
+Rp
+DS
+φ
+θ
+LOS/
+NLOS
+2
+MAE
+2.82
+0.58
+5.63
+4.16
+0.72
+99%
+STDE
+5.09
+1.93
+10.64
+9.63
+1.58
+N/A
+4
+MAE
+3.83
+1.01
+8.84
+7.03
+1.03
+95%
+STDE
+7.06
+3.08
+15.92
+15.13
+2.23
+N/A
+8
+MAE
+5.03
+1.58
+12.53
+10.65
+1.49
+90%
+STDE
+8.96
+4.29
+20.51
+20.18
+2.94
+N/A
+Table 3. Cumulative super-resolution performance of PL
+MAE
+STDE
+scale=2
+scale=4
+scale=8
+scale=2
+scale=4
+scale=8
++ATT
++5%
++3%
++6%
++6%
++4%
++5%
++DA
++16%
++12%
++12%
++10%
++5%
++6%
++RES
++11%
++7%
++9%
++11%
++5%
++6%
+STL
+0
+0
+0
+0
+0
+0
++ATT: Add attention mechanism to +DA.
++DA: Add data augmentation to +RES.
++RES: Add residual connection and iterative up-and-down to STL
+STL: The proposed model without RES, DA, and ATT in training.
+the fitting effect. After removing different strategies, we
+choose the most representative PL among the channel char-
+acteristics and test its performance with MAE and STDE.
+The results of the MAE and STDE ablation experiments are
+shown in Table 3, respectively. Here we use the model we
+designed as the baseline. RES represents the strategy of
+residual connections in the neural network, and the atten-
+tion mechanism will improve the super-resolution results
+by nearly 23%. DA will also have a significant effect on
+MAE. And it can be noticed that even with the increase of
+the super-resolution scale, the improvement by these meth-
+ods will still be stable.
+4
+Conclusion
+This paper proposes a residual-based SR model for wireless
+channel characteristics. We enhance the fitting ability of the
+proposed SR model by attention mechanism and generate a
+higher accuracy. A deep-shallow panel is used to expand
+the receptive field. We train our model using RT-urban con-
+structed by CloudRT platform. The proposed model can
+achieve SR performances of PL with MAE of 2.83db and
+99% accuracy of LOS areas given scale factor as 2. As
+the SR scale increases, this model maintains stable perfor-
+mance according to the numerical experiments. The pro-
+posed model are also compared with other state-of-the-art
+DL models such as ResNet, ViT, and GAN. In the future,
+we study the structure used in our current SR model to im-
+prove the accuracy of the SR of channel characteristics.
+Acknowledgements
+wait to add...
+References
+[1] C. Liu. Editorial: special topic on edge intelligence for
+internet of things. ZTE Communications, 19(2):01–01,
+2021.
+[2] Kapil Bhardwaj, Anant Singh, and Vibhav Kumar
+Sachan. 5g: An overview of channels characteristics
+and modelling techniques. In 2018 Fifth International
+Conference on Parallel, Distributed and Grid Comput-
+ing (PDGC), pages 400–405, 2018.
+[3] Sanaz
+Mohammadjafari,
+Sophie
+Roginsky,
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+Kavurmacioglu, Mucahit Cevik, Jonathan Ethier, and
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+[4] Danping He, Zhuocheng Xu, Huiyun Can, Yue Yin,
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+Vision – ECCV 2014, pages 184–199, 2014.
+[6] Yu Tian, Shuai Yuan, Weisheng Chen, and Naijin Liu.
+Transformer based radio map prediction model for
+dense urban environments. In 2021 13th International
+Symposium on Antennas, Propagation and EM Theory
+(ISAPE), volume Volume1, pages 1–3, 2021.
+[7] Lantu Guo, Yan Zhang, and Yue Li.
+An intelligent
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+work. Physical Communication, 44:101253, 2021.
+[8] Z. Zhang, X. Wang, D. He, Q. Huang, and D. Liu.
+Ray-tracing simulation and analysis of 5g channel char-
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+In 2022 IEEE In-
+ternational Symposium on Antennas and Propagation
+and USNC-URSI Radio Science Meeting (AP-S/URSI),
+pages 1690–1691, 2022.
+[9] Zhihao Wang, Jian Chen, and Steven C. H. Hoi. Deep
+learning for image super-resolution: A survey. IEEE
+Transactions on Pattern Analysis and Machine Intelli-
+gence, 43(10):3365–3387, 2021.
+This paper’s copyright is held by Haoyang Zhang. It is published in these proceedings and included in any archive such as
+IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of
+Scholarly Work.”
+

JtE3T4oBgHgl3EQfXgpN/content/tmp_files/load_file.txt

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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf,len=215
+page_content='URSI GASS 2023, Sapporo, Japan, 19 – 26 August 2023  Deep Learning Model with Attention Mechanism for Super-resolution of Wireless Channel  Characteristics  Haoyang Zhang(1), Xiping Wang (1), and Danping He(2)  (1) ABC University, Prague, Czechia;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' e-mail: FAA@seznam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='cz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' SBA@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='cz  (2) The Next University, Neverland, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' e-mail: TCA@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='com  Abstract  As an emerging approach, deep learning plays an increas-  ingly influential role in channel modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' In this paper, we  present a super-resolution (SR) model for channel charac-  teristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Residual connection and attention mechanism are  applied to this convolutional neural network (CNN) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  Experiments prove that the proposed model can reduce the  noise interference generated in the super-resolution process  of channel characteristics and reconstruct low-resolution  data with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The mean absolute error of our  channel SR model on the PL achieves the effect of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='82 db  with scale factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Compared with traditional ray tracing  methods and vision transformer (ViT), the proposed model  demonstrates less running time and computing cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  1  Introduction  With the rapid development of wireless communication  technology, 5G is widely used in various fields and daily  life applications, such as media streaming, gaming, and  video conferencing[1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Accurate channel model is regarded  as the foundation of future wireless network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Generally,  deterministic and semi-deterministic modeling are the two  mainstream methods of channel modeling[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  However,  this method has many limitations, including low computa-  tional efficiency, excessive computing power consumption,  and it requires complicated simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  Many researchers are trying to make breakthroughs in chan-  nel modeling by machine learning (ML)[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Because of its  generalizable architecture, machine learning is widely used  in almost every branch of science and technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Chan-  nel modeling is not an exception[4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' As for ML methods,  deep learning (DL) models of super-resolution (SR) such as  CNN [5], Transformer [6] and generative adversarial net-  work (GAN) [7] are frequently employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' However, the  scope of current research is still limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Most studies are  based on pure electromagnetic environment data without  considering complex terrain and building distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  In this paper, we propose a DL model of SR for channel  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  It recovers high-resolution (HR) charac-  teristics over low-resolution (LR) scenes built from origi-  nal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We transform the channel characteristics gener-  ation problem into a super-resolution problem of feature  maps containing various electromagnetic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' This  problem is addressed by using CNN with residual connec-  tion and attention mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The overview of our pro-  posed model is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We used CloudRT[8] for  ray tracing simulation, simulating the propagation of radio  waves in complex environments in dense urban areas, and  obtained channel characteristic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Besides mean  absolute error (MAE), we also incorporate the standard de-  viation error (STDE) into the loss function to balance reli-  ability and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We evaluate our proposed model by abla-  tion study and comparisons with other SR models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  2  Methodology  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='1  Data preprocessing and construction  Utilizing CloudRT, we developed a high-precision channel  feature dataset, RT-urban, which is used for training our  proposed SR model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Simulation configuration details are  summarized in[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We obtained seven channel characteris-  tics by simulation which are the height of buildings (h), path  loss (PL), multipath power ratio (Rp), and LOS/NLOS area  classification, root mean square (RMS) delay (DS), RMS  azimuth angle spread (φ), and RMS elevation angle spread  (θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='The latter six characteristics are our SR targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' 462 ray  data were generated in more than 70 dense urban areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  Values of channel characteristics that are far beyond ordi-  nary thresholds in communication systems are set as the  minimum (PL, Rp) or the maximum (DS) of the corre-  sponding normal range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' NaN value represents the chan-  nel characteristics data of receivers located inside buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  This kind of value should be void but set as a real number  out of the normal range so that the DL model can distin-  guish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The related data and processing methods are shown  in our previous work[8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='2  Neural network architecture  The super-resolution problem of the channel feature can be  expressed as a super-resolution problem of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The  key to image super-resolution lies in recovering from LR  data to HR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' [9]  This paper’s copyright is held by Haoyang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' It is published in these proceedings and included in any archive such as  IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of  Scholarly Work.”  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='04479v1  [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='SP]  11 Jan 2023  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The overview of proposed SR model for wireless  channel characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='1  Deep-shallow pipe based on residual network  In this study, we want to directionally generate various HR  channel features from the collected and processed LR chan-  nel feature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' There are differences between different  channel characters, and if all characters are fed into our  network, the results will be unsatisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  We propose  a method of classifying the propagation of deep-shallow  channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' 2, the deep-shallow backbone has a deep and shal-  low panel, which extracts features of different dimensions  from input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The deep panel has two convolutional lay-  ers with activation function ReLU more than the shallow  panel and repeats the illustrated convolution block multiple  times to extract features fully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The number of this block N  in our work is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We also introduce the method of residual  connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' It helps to expand the field of view and reduce  the loss of features caused by excessive convolution opera-  tions in the process of convolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  �  Hdeep = Fdeep(Iin,Wd)+Iin  Hshallow = Fshallow(Iin,Ws)+Iin  (1)  Where H is the output after convolution, Iin and W present  the input data and weights learn from deep and shallow pipe  F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' After each batch of convolution block, the unprocessed  input data is residually connected to the feature maps pro-  duced by the operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' It can help prevent gradient dis-  sipation during back-propagation, thus making it easier to  train deep networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='2  Attention mechanisms and feature blocks  It has to be considered that multi-path channel character-  istics such as PL, DL, and LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The learning patterns of  these features are completely different and tend to show dis-  tinguished differences in the learning process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' So we intro-  duced an attention mechanism for multi-feature extraction  on the framework of the SR model in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We split the  feature map F into several branches Klist by using kernels  with different sizes:  Klist = Conv(F,kernellist = [1,3,5,7]size)  (2)  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The overview of the proposed SR model with an  attention mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Conv(x) means that the convolution  scale is x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' PL* stands for PL, DS, and Rp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The overview of attention layer  As stated before, our goal is to control the information flows  from multiple characteristics carrying different scales of in-  formation into the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' To achieve this, we need to  integrate channels from the first N branches of the K list:  S = ∑  i  Ki,  i = [1,3,5,7]  (3)  then we embed the global information by simply using av-  erage pooling to generate channels as W ∈ RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The C chan-  nels are calculated by the following in dimensions h×w:  Wc = Favg(S) =  1  h×w  h  ∑  i  w  ∑  j  S(i, j)  (4)  So we can get a compact weight w ∈ R64×64 of different  channels by a simple fully connected layer, with the re-  duction of dimensionality for better efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' By fusing  branches of features, this mechanism enhances the ability  of the previous deep and shallow panels to extract char-  acteristics and it will also pay attention to multi-scales of  information when dell with different learning patterns of  characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='3  Loss functions  We generally use pixel loss instead of content loss for char-  acteristics learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Through experiments, we found  that the L1loss can help achieve better results on our task  than the peak signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  In data preparation,  we mentioned six different channel features in this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  However, the characteristic of LOS is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' It only has  This paper’s copyright is held by Haoyang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' It is published in these proceedings and included in any archive such as  IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of  Scholarly Work.”  DataSet  Loss function  high  Ray-Tracing  resolution(HR)  Validation  LosSi1(HR, lout)  Down  sampling  low  Input  SR Model  iteration  resolution(LR)  compare  evaluate  Deep -  Input  panel  Channel  multiply  attention  LR Data  blocks  Shallow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  Base Model  panelH  Nlos  Block  1  128  ATT  64  64  Deep feature panel  Theta  ayer  Block  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='.  (c, 200, 200)  Input  PL*  Block  Shallow feature panel  F(64, 200, 200)  Output  Fine-tune Part  Transfer  Reshape  Conv(x)  Relu  Block  Attention  Conv(7)  Conv(64)=wxK  m  Kernel 5x5  M  +  N blocks  F(64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=" 200)  S(64, 200, 200)  N blocks  F'(64, 200, 200)  N3  W(64, 1, 1  Kernel 7x7  Learn w E R64*64  from full connection  K(64, 200, 200)  K'(64, 200, 200)two types of integer values." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Therefore, the performance  can only be judged by valuing the accuracy of the model  for these two types of numerical classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' As a result  cross-entropy is used as the evaluation index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  lossl1(ˆI,I) =  n  (hw)2 ∑  i, j  | ˆIi, j −Ii, j|  (5)  lossce(ˆI,I) = −  n  (hw)2 ∑  i, j  Ii, j,klogˆIi, j,k  (6)  Where n,h,w represents the number of batches to be esti-  mated and the length and width of the region we chose for  input data, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' In order to enhance the confidence  of the fitted data, we also include the STDE as part of the  evaluation index of the fitting effect of the loss function:  STDE(ˆI,I) =  �  1  hw ∑  i, j  (ˆIi, j −Ii, j)2  (7)  The standard deviation reflects the degree of dispersion be-  tween the pixel value of the image and the mean value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The  smaller the standard deviation, the higher the fitting degree  of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  3  Experiment  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='1  Performance of the proposed model  (a) MAE of Path loss  (b)  Classification  accuracy  of  LOS/NLOS  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' MAE and classification accuracy of 2 main SR  targets during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  It can be observed from the experiment results that, the  MAE of our channel SR model on the PL achieves the effect  of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='82 db with scale factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Under the same conditions,  the comparison results of ResNet50, Vit, GAN, and UNet  are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' 5 and Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Obviously, after testing,  it can be found that these DL models, which achieved good  results in other fields such as computer vision or natural  language processing, performed worse than the proposed  model in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' On the single-task fitting for PL, the  performance of ResNet50 and UNet is around 7-8 db on  average after several epochs of training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Moreover, ViT can  only reach 8 db after modifying more layers and processing  with masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The best result of GAN is even larger than 12  db and still contains much noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Our model has achieved  results far exceeding the popular SR models on the channel  super-resolution task through the comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  On the RT-Urban dataset we constructed, we performed SR  training on 6 main channel features with scales of 2, 4, and  8, and the results are shown in Figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' 4 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Dur-  ing the training process of the shared parameter layer, it  can be found that the loss of each feature decreases rapidly  around the first 100 epochs and achieves a relatively sta-  ble result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The performance of the following targeted fea-  ture extractor can be optimized by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='5-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='7 db on the best  achieved by the backbone after fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  In the classification training of LOS/NLOS and θ, it can  be observed that the second half of SR training at scale 8  is more volatile and not as smooth as other features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' After  splitting the feature map data and visual analysis, we found  that large-scale downsampling will make the edge of the  classification area seriously jagged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Moreover, extracting a  more accurate mapping relationship is impossible and will  affect accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' However, the classification can still achieve  a correct rate of more than 91%, indicating that our model  has an imposing recovery effect on the channel feature data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  (a) Ground-Truth  (b) UNet  (c) ViT  (d) Ours  (e) GAN  (f) ResNet50  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Visualization of Super-Resolution results of DS  in a region of 1000 × 1000 m2 with scale factor 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Quantitative results for the experiments  Dataset  Method  MAE/db  STDE  RT-Urban  ViT  7-8  11-12  GAN  12-14  20-22  GANSR  7-8  17-20  ResNet50  6-7  9-10  UNet  6-7  12-14  Ours  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='83 (best)  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='09  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='2  Ablation study  The ablation experiments are used to verify whether these  methods we take to improve the SR of the model improve  This paper’s copyright is held by Haoyang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' It is published in these proceedings and included in any archive such as  IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of  Scholarly Work.”  8  Scale = 2  Scale = 4  7  Scale = 8  Path Loss (dB)  3  50  100  150  200  0  Epochs100  98  96  94  Accuracy (  92  90  88  86  Scale = 2  Scale = 4  84  Scale = 8  82  100  150  200  50  0  EpochsTable 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Super resolution performance (MAE & STDE)  Scale  Targets  PL  Rp  DS  φ  θ  LOS/  NLOS  2  MAE  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='82  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='58  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='63  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='16  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='72  99%  STDE  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='09  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='93  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='64  9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='63  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='58  N/A  4  MAE  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='83  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='01  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='84  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='03  95%  STDE  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='06  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='08  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='92  15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='13  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='23  N/A  8  MAE  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='03  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='58  12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='53  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='65  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='49  90%  STDE  8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='96  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='29  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='51  20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='18  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='94  N/A  Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Cumulative super-resolution performance of PL  MAE  STDE  scale=2  scale=4  scale=8  scale=2  scale=4  scale=8  +ATT  +5%  +3%  +6%  +6%  +4%  +5%  +DA  +16%  +12%  +12%  +10%  +5%  +6%  +RES  +11%  +7%  +9%  +11%  +5%  +6%  STL  0  0  0  0  0  0  +ATT: Add attention mechanism to +DA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  +DA: Add data augmentation to +RES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  +RES: Add residual connection and iterative up-and-down to STL  STL: The proposed model without RES, DA, and ATT in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  the fitting effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' After removing different strategies, we  choose the most representative PL among the channel char-  acteristics and test its performance with MAE and STDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  The results of the MAE and STDE ablation experiments are  shown in Table 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' Here we use the model we  designed as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' RES represents the strategy of  residual connections in the neural network, and the atten-  tion mechanism will improve the super-resolution results  by nearly 23%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' DA will also have a significant effect on  MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' And it can be noticed that even with the increase of  the super-resolution scale, the improvement by these meth-  ods will still be stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  4  Conclusion  This paper proposes a residual-based SR model for wireless  channel characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We enhance the fitting ability of the  proposed SR model by attention mechanism and generate a  higher accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' A deep-shallow panel is used to expand  the receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' We train our model using RT-urban con-  structed by CloudRT platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The proposed model can  achieve SR performances of PL with MAE of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='83db and  99% accuracy of LOS areas given scale factor as 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' As  the SR scale increases, this model maintains stable perfor-  mance according to the numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' The pro-  posed model are also compared with other state-of-the-art  DL models such as ResNet, ViT, and GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' In the future,  we study the structure used in our current SR model to im-  prove the accuracy of the SR of channel characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='  Acknowledgements  wait to add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
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+page_content='  This paper’s copyright is held by Haoyang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}
+page_content=' It is published in these proceedings and included in any archive such as  IEEE Xplore under the license granted by the “Agreement Granting URSI and IEICE Rights Related to Publication of  Scholarly Work.”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtE3T4oBgHgl3EQfXgpN/content/2301.04479v1.pdf'}

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+1 
+ 
+The guide to the guiding center aka pseudo-momentum operator 
+construction 
+E.L. Rumyantsev and A.V. Germanenko 
+School of Natural Science and Mathematics, Ural Federal University,  
+620002 Ekaterinburg, Russia 
+ 
+Abstract 
+The strictly gauge invariant approach to the construction of the analog of guiding center integrals 
+of motion in spatially homogeneous/inhomogeneous constant magnetic fields is considered. With 
+their help the gauge invariant equations, describing the wave functions of highly degenerate Lan-
+dau levels in the “classical” non-relativistic case, are formulated. The proposed gauge-invariant 
+approach was used also for the construction of the equations describing the quasi-relativistic car-
+riers’ behavior in the homogeneous /inhomogeneous magnetic field in the single layer graphene. 
+ 
+Keywords: Gauge invariance, vector potential, pseudo-momentum operator, guiding center oper-
+ator, graphene, SUSY equations, spatially homogeneous/inhomogeneous constant magnetic fields 
+1 Introduction 
+Guiding center approximation (or drift approximation) is a well-known and powerful theoretical 
+tool to describe the “classical” charge particle motion in plasma in a strong magnetic field [1]. This 
+most widely used approach allows to decouple fast helical motion of the particle about a local 
+magnetic line from the slow bounce and drift motions along and across magnetic field lines [2,3]. 
+The notion of the guiding center operator as the certain operator integration constant arises also in 
+the quantum mechanical description of the motion in a constant, spatially uniform magnetic field 
+[4,5,6]. In what follows we intend to propose the gauge invariant method of constructing the so 
+called pseudo-momentum operators which can be used for labelling wave functions of the highly 
+degenerate Landau levels and which are directly connected to the guiding center variables in their 
+classical meaning.   
+2 Uniform magnetic field problem revisited 
+The motion of a particle of mass 𝑚 and charge 𝑞 in uniform constant magnetic field is one of the 
+most studied quantum systems. Due to specific algebraic structure of the Hamiltonians considered, 
+as in relativistic case (Dirac Hamiltonian), so in non-relativistic case (Schrodinger Hamiltonian), 
+the energy spectrum can be easily obtained without turn to the solution of corresponding differen-
+
+2 
+ 
+tial equations. Nevertheless, to our point of view there are some questions to be clarified concern-
+ing the derivation of the Eigen wave functions in this seemingly simple and thoroughly scrutinized 
+problem. The problem hinges on the necessity to fix the form of the vector potential to achieve 
+this goal. Starting from the papers published by E. H. Kennard, C. C. Darwin and V. Fock [7,8,9] 
+it was common to use mainly circular gauge 𝑨 = [𝑩 × 𝒓] 2
+⁄ . For the beginning, we reconsider this 
+simplest case of non-relativistic 2D motion of the particle with the charge |𝑞| in the X-Y plane 
+perpendicular to uniform constant external magnetic field 𝐵 > 0 directed along Z-axis. To sim-
+plify our consideration, we neglect spin. The Hamiltonian to be considered in the first quantization 
+runs as  
+𝐻̂ = 𝑚
+2 (𝑣̂𝑥
+2 + 𝑣̂𝑦
+2)                             (1) 
+Where 𝒗 = (𝒑 − |𝑞|𝑨) 𝑚
+⁄
+, due to minimal coupling hypotheses. Hereafter ℏ = 𝑐 = 1. So defined 
+velocity component operators satisfy the following commutation rule [𝑣̂𝑥
+(+), 𝑣̂𝑦
+(+)] = 𝑖 𝑙𝐵
+2𝑚2
+⁄
+  𝑙𝐵 =
+√1 |𝑞|𝐵
+⁄
+ (henceforth we put ℏ = 𝑐 = 1). The index (+) is used to underline that we describe 
+motion of the particle with the charge |𝑞|.  The velocity operators can be redefined to reveal the 
+equivalence of the considered problem to 1D problem of harmonic oscillator. To this purpose, the 
+“quasi-position” 𝑄̂ = 𝑣̂𝑥
+(+)𝑚𝑙𝐵  
+2  and “quasi-momentum” 𝑃̂ = 𝑚𝑣̂𝑦
+(+)operators can be introduced 
+which fulfill the usual commutation rules [𝑄̂, 𝑃̂] = 𝑖 valid for the position and momentum opera-
+tors. The Hamiltonian in these operators is formally equivalent to the traditional 1D harmonic 
+oscillator 
+one. 
+This 
+redefinition 
+allows 
+also 
+to 
+construct 
+Bose 
+operators 
+𝑎̂ =
+𝑙𝐵𝑚(𝑣̂𝑥
+(+) + 𝑖𝑣̂𝑦
+(+)) √2
+⁄
+,  𝑎̂+ = 𝑙𝐵𝑚(𝑣̂𝑥
+(+) − 𝑖𝑣̂𝑦
+(+)) √2
+⁄
+ subjected to the commutation relation 
+[𝑎, 𝑎+] = 1, valid in considered case of constant spatially homogeneous magnetic field. The Ham-
+iltonian (1) in these operators acquires the form 𝐻̂ = 𝜔𝑐(𝑎̂+𝑎̂ + 1 2
+⁄ ) where 𝜔𝑐 = 𝑞𝐵 𝑚
+⁄
+. Choos-
+ing symmetric gauge 𝑨 = 𝐵 (−𝑦, 𝑥) 2
+⁄ , it is possible to introduce an additional pair of Bose oper-
+ators commuting with 𝑎̂ and 𝑎̂+ by simple changing the sign of the charge and interchanging an-
+nihilation/creation operators [5,10] 
+𝑏̂+ = 𝑙𝐵��
+√2
+(𝑣̂𝑥
+(−) + 𝑖𝑣̂𝑦
+(−)) = 𝑙𝐵
+√2
+[(𝑝𝑥 + |𝑞|𝐴𝑥) + 𝑖(𝑝𝑦 + |𝑞|𝐴𝑦)]       (2) 
+𝑏̂ = 𝑙𝐵𝑚
+√2
+(𝑣̂𝑥
+(−) − 𝑖𝑣̂𝑦
+(−)) = 𝑙𝐵
+√2
+[(𝑝𝑥 + |𝑞|𝐴𝑥) − 𝑖(𝑝𝑦 + |𝑞|𝐴𝑦)] 
+It is common to define with the help of these operators the coordinates of the center of circular 
+orbit along which the charged particle is gyrating (guiding center operators) [5]. It must be stressed 
+that so written expressions for 𝑏̂, 𝑏̂+ are misleading. If we assume that 𝐴𝑖 in (2) are really the 
+
+3 
+ 
+components of a vector potential, we are to accept that the gauge invariance of our solutions is 
+violated.  Really, 𝑣̂𝑖
+(−) in this case are to be identified with the velocity operators for the particle 
+with the charge - |𝑞| (positron?!) which cannot appear in our non-relativistic theory. Moreover, 
+the straightforward evaluation e.g. of the commutator [𝑎̂, 𝑏̂+] leads to the following condition to 
+be imposed on chosen gauge 
+[𝑎̂, 𝑏̂+] = −𝑖𝑙𝐵
+2|𝑞|[𝜕𝑥𝐴𝑥 − 𝜕𝑦𝐴𝑦] + 𝑙𝐵
+2|𝑞|[𝜕𝑦𝐴𝑥 + 𝜕𝑥𝐴𝑦]   (3) 
+Which is zero for the symmetric gauge 𝑨 = 𝐵(−𝑦, 𝑥) 2 
+⁄
+ only . So, we are to write e.g. operator 
+𝑏̂ as 
+𝑏̂ = 𝑙𝐵
+√2
+[(𝑝𝑥 + |𝑞|𝐴̃𝑥) − 𝑖(𝑝𝑦 + |𝑞|𝐴̃𝑦)]                (4) 
+Where 𝐴̃𝑖 are the components of some vector field, determined in “fixed” symmetric gauge as 
+𝐴̃𝑖 = −𝐴𝑖. In order to lend support to this statement let us consider the problem of guiding center 
+operators from another point of view which has been discussed e.g. in [11]. We start from classical 
+description where it is possible to solve the problem of the motion in constant magnetic field em-
+ploying extra conserved quantity 𝒌 [12,13]. This vector emerges in classical description when we 
+integrate the equation of the motion 𝑚 𝑑𝒗 𝑑𝑡
+⁄
+= |𝑞|𝒗 × 𝑩 with respect to time, with the result 
+𝑚𝒗 = |𝑞|𝒓 × 𝑩 + 𝒌. The meaning of the integration constant 𝒌 = 𝑚𝒗 − |𝑞|𝒓 × 𝑩 is clarified af-
+ter proper scaling and rotation [13,14,15]. The vector 𝑹0 = [𝒌 × 𝑩] |𝑞|𝐵2
+⁄
+ in the classical picture 
+defines the center of particle circular motion (guiding center) fixed at the moment of magnetic 
+field switching on. It has to be mentioned that this integral of motion has been established by 
+Gorkov and Dzyaloshinskii in [16], see also [17]. It is easy to verify that in quantum mechanical 
+description 𝒌̂ (now 𝒌 becomes an operator) remains also time-invariant, as  
+𝜕𝒌̂
+𝜕𝑡 = 𝑖[𝐻̂, 𝒌̂] = 𝑖 [𝑚(𝑣̂𝑥2 + 𝑣̂𝑦2)
+2
+, 𝑚𝒗̂ − |𝑞|[𝒓̂ × 𝑩]] = 0          (5) 
+The introduced 𝒌̂ operators are subjected to more strict commutation conditions in considered 
+uniform magnetic field case namely [𝑘̂𝑖, 𝑣̂𝑗] ≡ 0 regardless of the specific choice of the vector 
+potential. This property will be of use for us later on while discussing the graphene behavior under 
+the action of the spatially homogeneous/inhomogeneous constant magnetic fields. The explicit 
+forms of these operators run as follows 
+𝑘̂𝑥 = 𝑚𝑣̂𝑥 − 𝑦
+𝑙𝐵
+2   𝑘̂𝑦 = 𝑚𝑣̂𝑦 + 𝑥
+𝑙𝐵
+2          (6)  
+Pay attention that contrary to the statement in [18], these operators are strictly gauge invariant. 
+Really, the physical meaning of the terms |𝑞|[𝒓̂ × 𝑩] after scaling and rotation (as in the case of 𝒌̂ 
+
+4 
+ 
+) is revealed as the particle coordinates operators in disguise, which of course do not respond to 
+the gauge transformations and the velocity operators 𝑣̂𝑖 are gauge invariant by the definition.  As 
+𝒌̂ has dimension of momentum, this vector constant is known under the term “pseudo-momentum” 
+(the nomenclature used in [18,19]). As components of 𝒌̂  do not commute ([𝑘𝑦, 𝑘𝑥] = 𝑖 𝑙𝐵  
+2 )
+⁄
+ but 
+nevertheless commute separately with the Hamiltonian, it is useful to construct from them the 
+ladder operators  
+                                 𝑏̃ = 𝑙𝐵 (𝑘𝑦 + 𝑖𝑘𝑥) √2
+⁄
+     𝑏̃+ = 𝑙𝐵 (𝑘𝑦 − 𝑖𝑘𝑥) √2
+⁄
+                     (7) 
+ 
+In order to bring them both simultaneously into play for labelling the eigenfunction. These oper-
+ators are in one-to-one correspondence with the pair of 𝑏-operators used in [5,10] and discussed 
+above, which practically in all the papers known to us, are used for the construction of the “cen-
+ter of rotation” operators. We can require the eigenfunction Ψ𝑛,𝜆(𝒓)belonging to the given n’th 
+Landau level to be simultaneously the eigenfunction of  𝑏̃  
+𝑏̃ Ψ𝑛,𝜆(𝒓) = 𝜆Ψ𝑛,𝜆(𝒓)                (8) 
+where 𝜆 arbitrary complex number 𝜆 = 𝜆1 + 𝑖𝜆2 (the celebrated coherent states). Or we can use 
+operator  𝑏̃+𝑏̃ (𝑚 > 0) 
+𝑏̃+𝑏̃Ψ𝑛,𝑚(𝒓) = 𝑚Ψ𝑛,𝑚(𝒓)             (9) 
+As this second variant describe the states which are all gyrating about fixed center 𝒓 = 0, they 
+contradict our classical picture. We with the authors [5] adhere to the first choice which has simple 
+and physically clear interpretation and coincides with the classical description. The second variant 
+can be of use for the description of the charge particle motion in an axisymmetric magnetic field 
+with straight field lines dependent only on |𝒓| [20].   One more comment is due. The defined 
+coherent states formed non-orthogonal over complete set for arbitrary 𝜆. It is known that this set 
+can be reduced to orthogonal complete set on the von Neumann lattice [21]. One more possibility 
+to use both non-commuting pseudo-momentum operators simultaneously arises when we try to 
+impose periodic boundary conditions following [10,22] with the help of the shifting operators 𝑇̂𝑥 =
+exp (𝑖𝑘̂𝑥𝑥) and 𝑇̂𝑦 = exp (𝑖𝑘̂𝑦𝐿𝑦). 𝐿𝑥 and 𝐿𝑦 define a parallelogram where the particle resides. 
+The periodic conditions demand that 
+𝑇̂𝑥Ψ = 𝑒𝑖𝜃𝑥Ψ    𝑇̂𝑦Ψ = 𝑒𝑖𝜃𝑦Ψ         (10) 
+ These conditions can be fulfilled if and only if [𝑇̂𝑥, 𝑇̂𝑦] = 0 which due to [[𝑘𝑦, 𝑘𝑥] = 𝑖 𝑙𝐵  
+2
+⁄
+im-
+pose restrictions on the choice of 𝐿𝑥 and 𝐿𝑦 
+𝐿𝑥𝐿𝑦 
+𝑙𝐵  
+2
+= 2𝜋𝑛      (11) 
+
+5 
+ 
+Where 𝑛 is any integer number.   
+One more essential for the gauge invariance of 𝑘̂𝑖-operators property is revealed, if we present 
+them in arbitrary gauge 𝑨  in the form (𝑚𝒗̂ = 𝒑̂ − |𝑞|𝑨) 
+𝒌̂ = 𝒑̂ − 1
+2 |𝑞|[𝒓̂ × 𝑩] − |𝑞| (𝑨 + 1
+2 [𝒓̂ × 𝑩])           (12)         
+It is easy to verify that the combination in parentheses is curl independent, thus meaning that in-
+tegral 
+∫ (𝑨 + 1
+2 [𝒓̂ × 𝑩]) 𝑑𝒓
+𝒓2
+𝒓1
+    (13) 
+Is path independent. Thus, we can associate with this expression the gradient of some function, 
+which can be dubbed as generalized gradient transformation function 
+𝑨 + 1
+2 [𝒓̂ × 𝑩] = 𝛁𝜑     (14) 
+This well-known combination has appeared in the famous paper by J. Schwinger [23] presenting 
+derivation of relativistic electron propagator within original essentially gauge invariant method. It 
+has been shown that this method can be applied for computing non-relativistic propagator as well, 
+though unfortunately this method is rarely used in this context [24,25]. This curl vanishing expres-
+sion appeared in [23] in the relativistic invariant form 𝐴𝜇(𝑥) + 𝐹𝜇𝜈(𝑥) 2
+⁄  where 𝐹𝜇𝜈 = 𝜕𝜇𝐴𝜈 −
+𝜕𝜈𝐴𝜇. It is easy to check that in our 2D non-relativistic case this expression coincides with (13). 
+Inasmuch according to the commonly accepted prescription gauge transformation function 𝜑 has 
+no any  effect upon wave function other than multiplication by phase factor 𝑒𝑥𝑝𝑖𝑞𝜑, and corre-
+spondently (as it is prescribed) can be ignored, we are left with the expression 𝒌̂ = 𝒑̂ −
+1
+2 |𝑞|[𝒓̂ × 𝑩].  The subtle point is that we cannot identify [𝒓̂ × 𝑩] 2
+⁄  with the vector potential – 𝑨, 
+as in this case 𝒌̂ would be gauge dependent quantity as it was clarified above. The operator 𝒌̂ =
+𝒑̂ + |𝑞|𝑨 so understood is the mechanical moment for the “anti-particle” and according to charge 
+super-selection rule would be acting in orthogonal Hilbert subspace [26,27]. Such difference in 
+the behavior of these two terms of the pseudo-momentum under gauge transformation is reminis-
+cent of the point of view presented in [28,29,30]. The only difference is that the authors propose 
+to consider peculiarities not in 𝒌̂  transformation but in the redefined vector potential   𝑨 =
+𝑨𝑝ℎ𝑦𝑠 + 𝑨𝑝𝑢𝑟𝑒. Their basic postulate is that under gauge transformation 𝑈 = exp (𝑖𝑞𝜒), these two 
+components transform differently. 𝑨𝑝𝑢𝑟𝑒 transforms as the full 𝑨 (𝑨𝑝𝑢𝑟𝑒 → 𝑨𝑝𝑢𝑟𝑒 + ∇𝜒), while 
+
+6 
+ 
+𝑨𝑝ℎ𝑦𝑠 transforms in the same manner as does the electric field 𝑬 thus remaining unchanged  
+𝑨𝑝ℎ𝑦𝑠 → 𝑨𝑝ℎ𝑦𝑠 (see [30] and citing within).  
+The well-studied problem in homogeneous field nevertheless arises two questions. First, if 
+we discard defined above 𝛁𝜑 in 𝒌̂  through gauge transformation of the wave function Ψ(𝒓) =
+Φ(𝒓)exp 𝑖|𝑞|𝜑,  the equation for Φ(𝒓) will contain “fixed”  symmetric vector potential [𝑩 × 𝒓̂] 2
+⁄  
+independently of the form of our initially arbitrary chosen potential 𝑨 . Such property of the con-
+sidered approach resolves long-standing puzzle of the linear Landau gauge 𝑨1 = 𝐵(0, 𝑥)or  𝑨2 =
+𝐵(−𝑦, 0). Contrary to common statement, if we rely on the “symmetry” considerations and an-
+nounce the eigenfunctions to be of the form 𝜑(𝑥)exp(𝑖𝛾𝑦) for 𝑨1 (𝜑(𝑦)exp(𝑖𝛾𝑥) for 𝑨2), the 
+obtained wave functions in these gauges does not belong to the space of square integrable functions 
+of the symmetric gauge thus contradicting announced gauge invariance. The problem of the map-
+ping states in 𝑨1 gauge to the states in 𝑨2 gauge is also not so simple and straightforward as has 
+been clarified in [31]. The outlined approach show, that due to the existence of guiding center 
+integral of motion, starting with arbitrary 𝑨, we arrive at the equations written in the symmetric 
+gauge which lead uniquely to the solutions with the finite norm. It must be noted that the problem 
+of such possible “uniqueness” of the vector potential choice has been discussed albeit from another 
+point of view in [32,33].  
+ 
+3 Non-relativistic particle in the inhomogeneous field 
+Following the approach outlined above, we present below the construction of an analog of the 
+guiding center operator for spatially inhomogeneous magnetic field [34]. We choose e.g. a mag-
+netic field with a constant gradient given by  
+𝑩(𝒓) = 𝐵0
+𝑥
+𝐿 𝒛 ̂             (15) 
+ Where 𝐵0 is a constant and 𝐿 ≡ |∇ln (𝐵)|−1 is the constant gradient length scale. This seemingly 
+oversimplified example of spatially inhomogeneous field is nevertheless important for the descrip-
+tion of the charge particle motion in the region of magnetic field reversal leading to formation of 
+current sheets along neutral lines [35,36]. The classical variant of this problem has been discussed 
+in [37,38]. Being inspired by the form of guiding center operators in homogeneous case, we pro-
+posed that an analog of gauge invariant pseudo-momentum operator (if exists) in such field is also 
+of the form 
+𝒌̂ = (𝑚𝒗̂ − |𝑞|𝑨̃)            (16) 
+From now on we will use only gauge invariant velocity operators 𝒗 = (𝒑 − |𝑞|𝑨) 𝑚
+⁄
+, so the (+) 
+index will be omitted. It should be reminded that in no way 𝑨̃(𝒓) in the expression for 𝒌̂ can be 
+
+7 
+ 
+considered as the vector potential and as so it must remain unchanged under gauge transformation. 
+This vector field is to  be determined from the condition that 𝒌̂ (or its components) commute with 
+the Hamiltonian . This condition is a fortiori fulfilled if as discussed above 
+                                                   [𝑚𝑣̂𝑖, 𝑘̂𝑗] ≡ 0 𝑖, 𝑗 = 1,2            (17)  
+ 
+Taking into account that  [𝑣̂𝑥, 𝑣̂𝑦] = 𝑖|𝑞|𝐵(𝒓) 𝑚2
+⁄
+, these conditions lead to the following set of 
+equations  
+𝜕𝑥𝐴̃𝑥 = 0  𝜕𝑦𝐴̃𝑥 = 𝐵(𝒓)                   (18) 
+  𝜕𝑥𝐴̃𝑦 = −𝐵(𝒓)   𝜕𝑦𝐴̃𝑦 = 0                          
+Substituting chosen 𝐵(𝒓) = 𝐵(𝑥), we infer that only  𝐴̃𝑦 component complies with the required 
+conditions and is  
+𝐴̃𝑦 = −𝐵0
+𝑥2
+2𝐿                     (19) 
+By the way, the arising of such constant of motion (conserved quantity) can be inferred from the 
+classical equation of motion: 
+𝑣̇𝑥 =
+|𝑞|
+𝑚 𝐵0
+𝑥
+𝐿 𝑣𝑦     𝑣̇𝑦 = −
+|𝑞|
+𝑚 𝐵0
+𝑥
+𝐿 𝑣𝑥         (20) 
+Integrating the second equation we obtain 
+𝑣𝑦 = −
+|𝑞|
+𝑚 𝐵0
+𝑥2
+2𝐿 + 𝑘𝑦
+𝑚                       (21) 
+It is easy to verify that going to quantum mechanical description the so defined operator 𝑘𝑦 is the 
+required additional integral of motion. As we have only one conserved component of pseudo-
+momentum we are left with no choice but to state that the Eigen functions of the problem 
+𝐻̂Ψ𝐸(𝒓) = 𝐸Ψ𝐸(𝒓) are simultaneously the Eigen functions of found pseudo-momentum compo-
+nent 
+𝑘̂𝑦Ψ𝐸,𝜆(𝒓) = (−𝑖𝜕𝑦 − |𝑞|𝐴𝑦 − |𝑞|𝐴̃𝑦)Ψ𝐸,𝜆(𝒓) = 𝜆Ψ𝐸,𝜆(𝒓)            (22) 
+The vector potential 𝐴𝑦 in this expression is any “real” vector potential suitable to our problem, 
+which changes under gauge 𝑈(1) transformation. Once more we want to call the reader's  attention 
+to the specific behavior of 𝑘̂𝑦 operator under gauge transformation. Only 𝐴𝑦 term in this expression 
+undergoes change with gauge variation. The term 𝐴̃𝑦 is not affected by this operation as it is in 
+essence the function of particle coordinates and thus is not subjected to gauge transformations. 
+
+8 
+ 
+Now we are going to prove that notwithstanding the specific choice of the vector potential, ob-
+tained solutions belong to the same Hilbert space. Really, let us choose the vector potential in the 
+Landau-like (linear) gauge  𝑨 = 𝐵0(0, 𝑥2 2𝐿
+⁄
+). In this gauge 𝑘̂𝑦 = −𝑖𝜕𝑦 and thus the 𝑦 component 
+of the canonical momentum is conserved in accord with the commonly accepted approach based 
+on the symmetry of the problem [39]. In this case Ψ𝐸,𝜆(𝒓) = 𝜑𝐸,𝜆(𝑥)𝑒𝑥𝑝𝑖𝜆𝑦, where 𝜑𝐸,𝜆(𝑥) sat-
+isfies the equation (𝑙𝐵0
+2 = 1 |𝑞|
+⁄
+𝐵0) 
+                                 [−𝜕𝑥
+2 + (𝜆 − 𝑥2 2𝐿𝑙𝐵0
+2
+⁄
+)
+2] 𝜑𝐸,𝜆(𝑥) = 2𝑚𝐸𝜑𝐸,𝜆(𝑥)              (23)   
+ 
+Now, let us try the symmetry-like gauge in this problem, using for this purpose the so called 
+Poincare’ or multipole gauge (PMG) [40,41,42,43,44]. In its relativistic covariant form PMG po-
+tential satisfies condition 𝑥𝜇𝐴𝜇(𝑥) = 0. In this case it can be expressed as the integral over the 
+electromagnetic field tensor 𝐹𝜇𝜆(𝑥) = 𝜕𝜇𝐴𝜆(𝑥) − 𝜕𝜆𝐴𝜇(𝑥) as [45]  
+𝐴𝜇(𝑥) = ∫ 𝑑𝑢 𝑢𝑥𝜆
+1
+0
+𝐹𝜆𝜇(𝑢𝑥)           (24) 
+ In the considered by us non-relativistic limit this expression transforms into  
+𝑨(𝒓) = −𝒓 × ∫ 𝑑𝑢 𝑢
+1
+0
+𝐵(𝑢𝒓)𝒛̂              (25) 
+It is easy to verify that for spatially homogeneous field we obtain well-known potential in sym-
+metric gauge. In chosen magnetic field with constant gradient, according to (25), the chosen “ar-
+bitrary” vector potential is 
+𝑨(𝒓) = 𝐵0(−𝑦, 𝑥) 𝑥
+3𝐿            (26)    
+Correspondently, 𝑘̂𝑦 = −𝑖𝜕𝑦 + 𝑥2 6𝐿𝑙𝐵0
+2
+⁄
+ and Ψ𝐸,𝜆(𝒓) = 𝜑𝐸,𝜆(𝑥)exp [𝑖𝑦(𝜆 − 𝑥2 6𝐿𝑙𝐵0
+2
+⁄
+)]. It is 
+easy to verify that 𝜑𝐸,𝜆(𝑥) in this expression coincides with the one in (22), as it satisfies the same 
+equation. Thus it is proved that as in the case of the uniform magnetic field, so in our non-uniform 
+problem the requirement on the wave function to be the eigenfunction of the pseudo-momentum 
+leads to the one-to-one, up to the exponential phase factor, mapping of the solutions belonging to 
+the different gauges. It is interesting to compare our quantum mechanical problem with classical 
+solutions in constant gradient field discussed in [46, 47, 36]. Compare the expression for conserved 
+𝑌 component of the canonical momentum 𝑝𝑦 = 𝑚𝑣𝑦 + 𝑞𝐴𝑦 = 𝑐𝑜𝑛𝑠𝑡 (Formula (1) in [36]) which 
+is in one-to-one correspondence with considered guiding center operator 𝑘̂𝑦. It must be noted that 
+
+9 
+ 
+the quantum variant of this problem is by far more rich in physics as compared to its classical 
+counterpart. The quantum description for 𝜆 > 0 is given by 1D Schrodinger equation (23) with the 
+“celebrated quartic double-well potential” [48] which is omnipresent in different physical and 
+chemistry problems. The quantum solution  differs due to the tunneling effect (“instanton” behav-
+ior) from classical prediction of existing for 𝜆 > 0  localized “one-sided” gyration [36] which does 
+not cross 𝐵 = 0 line. Thus, as for 𝜆 > 0 so for 𝜆 < 0, the average particle quantum motion is 
+symmetric relative to neutral magnetic line. It must be stressed than despite the apparent differ-
+ences in the exponential factors of the considered wave functions the 𝑦 component of the physi-
+cally meaningful quantity –the current density remains invariant (as it must be!) under the gauge 
+change. Using the definition of the current density in magnetic field [Landau] it is straightforward 
+to show that in the both gauges the current density is 
+𝐽𝑦 = 𝑖𝑞
+2𝑚 [(𝜕𝑦Ψ∗)Ψ − Ψ∗𝜕𝑦Ψ] − 𝑞2
+𝑚 𝐴𝑦Ψ∗Ψ = |𝑞|
+𝑚 (𝜆 −
+𝑥2
+2𝐿𝑙𝐵0
+2 ) |𝜑(𝑥)|2       (27) 
+Pay attention that according to this expression, 𝜆 sign alone does not determine the direction of 
+the particle drift in the considered state. For 𝜆 > 0 all depends on the average mean-square value 
+of the particle deviation along 𝑋 axes 〈𝑥2 2𝐿𝑙𝐵0
+2
+⁄
+〉. For 𝜆 < 〈𝑥2 2𝐿𝑙𝐵0
+2
+⁄
+〉 the particle changes the 
+drift direction. This result is in accord with the classical considerations [36] and at the same time 
+reveals peculiar status of 𝒌̂ operators. The physical meaning as an observable must be ascribed 
+without doubt to 𝐽𝑦, which means that 𝑘̂𝑦 plays some auxiliary role and its consideration as ob-
+servable is under question. Here it is appropriate to remember (see above) that this characteristic 
+emerges in classic picture not as the constant of motion in its accepted meaning but as the constant 
+of integration over time. This suspicion of the strange role of the guiding center operator in our 
+quantum problem is reinforced by the revision of the solution in homogeneous magnetic field 
+discussed above. The states belonging to, e.g., ground Landau level (from which all others n-levels 
+wave functions can be deduced) are given by the equation 
+𝑎̂Ψ0(𝒓) = 𝑙𝐵𝑚
+√2
+(𝑣̂𝑥 + 𝑖𝑣̂𝑦)Ψ0(𝒓) = 1
+√2
+(𝑎̂𝑥 + 𝑖𝑎̂𝑦)Ψ0(𝒓) = 0        (28) 
+Where 𝑎̂𝑖, 𝑎𝑖
++ are the Bose operators [𝑎̂𝑖, 𝑎𝑗
++] = 𝛿𝑖𝑗 of the form  
+𝑎̂𝑖 = −𝑖𝑙𝐵 (𝜕𝑥𝑖 + 𝑥𝑖
+2𝑙𝐵
+2)   𝑎𝑗
++ = −𝑖𝑙𝐵 (𝜕𝑥𝑖 − 𝑥𝑖
+2𝑙𝐵
+2)          (29) 
+The general solution Ψ0,𝜆(𝒓) = φ0,𝜆(𝑥)φ0,𝑖𝜆(𝑦) of the equation () is given by the solutions of two 
+equations 
+
+10 
+ 
+𝑎̂𝑥φ0,𝜆(𝑥) =  𝜆φ0,𝜆(𝑥)    𝑎̂𝑦φ0,𝑖𝜆(𝑦) = 𝑖𝜆φ0,𝑖𝜆(𝑦)        (30)    
+Where φ0,𝜆(𝑥), φ0,𝑖𝜆(𝑦) are corresponding coherent states and 𝜆 = 𝜆1 + 𝑖𝜆2 arbitrary complex 
+number. So we obtain the highly degenerate set of the wave functions belonging to the same Lan-
+dau level without invoking the guiding center operators. Thus the use of pseudo-momentum oper-
+ators in this problem is superfluous. They can be used for clarifying the physical meaning of the 
+numbers 𝜆1,2 The action of the introduced above operator       𝑏̃ = 𝑙𝐵 (𝑘𝑦 + 𝑖𝑘𝑥) √2
+⁄
+ where {𝑘𝑖} 
+are the discussed pseudo-momentum operators upon these functions is 
+𝑏̃Ψ0,𝜆(𝒓) = 1
+√2
+ (𝑎𝑦 + 𝑖𝑎𝑥)Ψ0,𝜆(𝒓) = 𝑖𝜆√2                   (31) 
+So we can state that really the numbers𝜆1,2 labelling the eigenfunctions can be interpreted after 
+scaling by   𝜆𝐵 as the corresponding 𝑅2,1-coordinates of the center of the particle gyration. The 
+usefulness of these operators lies in the fact that with them we can construct gauge invariant equa-
+tions choosing for the start any appropriate gauge as it was clarified above. 
+ 
+4 Graphene in the magnetic fields 
+An additional but no less important example of proposed approach is due to the fact that introduced 
+above gauge invariant pseudo-momentum operators  𝑘̂𝑥, 𝑘̂𝑦 (6) remain valid as the motion con-
+stants for the description of the low-energy envelope states in the single layer graphene in homo-
+geneous field and defined above 𝑘̂𝑦 (20,21) can be used for labeling states in the perpendicular 
+gradient magnetic field 𝑩(𝒓) = 𝐵0𝑥 𝒛̂ 𝐿
+⁄ . Due to commutation conditions defined in (5) for ho-
+mogeneous field and   𝑘̂𝑦 commutator (17) in the gradient field case these operators commute with 
+the Dirac-like Hamiltonian  describing carriers behavior in graphene within 𝒌 ∙ 𝒑 approach and 
+can serve as the corresponding quantum numbers  [49,50]. Due to valley degeneracy of graphene 
+Hamiltonian valid for arbitrary perpendicular magnetic field it suffices as it is common to restrict 
+our consideration to one of the valleys (say K valley) described by the Hamiltonian 𝐻̂ =
+𝑣𝐹(𝑄̂+𝜎+ + 𝑄̂𝜎−) [51]. Here 𝑄̂ = 𝜋̂𝑥 + 𝑖𝜋̂𝑦, 𝜎± = (𝜎𝑥 ± 𝑖𝜎𝑦) 2
+⁄ , and 𝜎𝑖 are Pauli matrixes. Con-
+sider the behavior of the zero-mode states (if existing) described by the first-order partial differ-
+ential equation 
+(𝜋̂𝑥 + 𝑖𝜋̂𝑦)Ψ(𝒓) = 0           (32) 
+ Where 𝜋̂𝑖 = −𝑖𝜕𝑖 − |𝑞|𝐴𝑖(𝒓). It is straightforward to show that the sought-for solutions form the 
+set of coherent states being in one-to-one correspondence with the set describing the degenerate 
+lowest Landau level in the “classical” non-relativistic problem [see (28,29,30)]. As discussed 
+
+11 
+ 
+above, in the gradient magnetic field 𝑩(𝒓) = 𝐵0 𝑥𝒛̂ 𝐿
+⁄  we can choose any appropriate gauge. In 
+deciding on the gauge 𝑨 = (0, 𝐵0 𝑥2 2𝐿)
+⁄
+ we arrive to the simplest form for 𝑘̂𝑦 = −𝑖𝜕𝑦 (see dis-
+cussion above). Labeling the Eigen functions Ψ𝜆
+𝑇(𝒓) = (𝜑𝜆(𝑥), 0)exp(𝑖𝜆𝑦) by its eigenvalue 𝜆  
+we obtain  
+[−𝑖𝜕𝑥 + 𝑖(𝜆 − 𝑥2 2𝐿𝑙𝐵0
+2
+⁄
+)]𝜑𝜆(𝑥) = 0        (33) 
+It follows from (33) that 𝜑𝜆(𝑥)~exp(𝜆𝑥 − 𝑥3 6𝐿𝑙𝐵0
+2
+⁄
+). Contrary to the behavior in the considered 
+above non relativistic Schrodinger case  where a particle can remain localized along neutral line 
+[36] crossing it hither and thither, the zero-mode carriers in K valley escape to  𝑥 = −∞ thus 
+destroying the current sheet. The general Landau state (𝐸 ≠ 0) is given by the solution Φ(𝒓)𝑇 =
+(𝜑1(𝒓), 𝜑2(𝒓)) of the matrix equation 
+(−𝐸𝐼 + 𝑣𝐹𝑄̂+𝜎+ + 𝑣𝐹𝑄̂𝜎−)Φ(𝒓) = 0        (34) 
+These equations of the first order can be transformed to the equations of the second order by ap-
+plying to the equation (34) the operator  𝐸𝐼 + 𝑣𝐹𝑄̂+𝜎+ + 𝑣𝐹𝑄̂𝜎− 
+(𝐸𝐼 + 𝑣𝐹𝑄̂+𝜎+ + 𝑣𝐹𝑄̂𝜎−)(−𝐸𝐼 + 𝑣𝐹𝑄̂+𝜎+ + 𝑣𝐹𝑄̂𝜎−)Φ(𝒓) = 0        (35) 
+As a result, we are to solve two Schrodinger-like equations  
+(−𝐸2 + 𝑣𝐹
+2𝑄̂+𝑄̂)𝜓1(𝒓) = 0  (−𝐸2 + 𝑣𝐹
+2𝑄̂𝑄̂+) 𝜓2(𝒓) = 0          (36) 
+Which are super-symmetry (SUSY) connected [52], [53] as the solutions Ψ𝑇(𝒓) =
+(𝜓1(𝒓), 𝜓2(𝒓)) are subjected to the condition 𝜓2(𝒓) = 𝑣𝐹𝑄̂𝜓1(𝒓) 𝐸
+⁄ . As it has been clarified in 
+[54] the arbitrary solutions of these squared Dirac-like equations being of the second order can 
+contain “superfluous” ones which do not satisfy the initial equation of the first order. The remedy 
+is to consider the function  
+Φ(𝒓) = (𝐸𝐼 + 𝑣𝐹𝑄̂+𝜎+ + 𝑣𝐹𝑄̂𝜎−)Ψ(𝒓)         (37) 
+Which is the solution of the first order equation (34) if Ψ(𝒓) is the solution of (35). As we are left 
+with the solution of the one Schrodinger-like equation (36), the procedure outlined above for non-
+relativistic problem can be at once applied for the analysis of carrier spectrum in graphene. For the 
+chosen gradient magnetic field, the explicit form of the corresponding Schrodinger –like operator 
+is 
+𝑣𝐹
+2𝑄̂+𝑄̂ = 𝑣𝐹
+2(𝜋̂𝑥 − 𝑖𝜋̂𝑦)(𝜋̂𝑥 + 𝑖𝜋̂𝑦) = 𝑣𝐹
+2[𝜋̂𝑥2 + 𝜋̂𝑦2 − 𝑥 𝐿𝑙𝐵0
+2 ]
+⁄
+         (38) 
+The difference with the non-relativistic case discussed above resides in the linear in 𝑥 term (com-
+pare with (23)). The wave function solutions as in non-relativistic case demonstrate two types 
+
+12 
+ 
+depending on the sign of 𝜆, described above. The only difference with classical result is that they 
+are shifted in the positive direction along 𝑋 axis. The symmetry is restored when we consider 
+carrier behavior in 𝐾′valley. We will not proceed further with the analysis of the wave function 
+solutions and energy spectrum of (35) which will be considered elsewhere, as our task in the pre-
+sented paper has been to pave gradient invariant road to the formulation of “proper” wave equa-
+tions with the help of pseudo-momentum aka guiding center operator.    
+5 Conclusion 
+The role of the “forgotten” pseudo-momentum in the solution of the Landau problem as for uni-
+form so in spatially non-uniform magnetic fields has been discussed in the series of the papers (see 
+[11] and citing herein). Presented approach differs in that we placed particular emphasis on the 
+gauge invariance of the procedure of the construction of the corresponding wave equations. It is 
+common consensus in that the gauge invariance is one of the most fundamental symmetry proper-
+ties of physics [55,56]. Thus, citing J. Schwinger [23], we follow his prescription that “a formally 
+gauge invariant theory is ensured if one employs methods of solution that involve only gauge 
+covariant quantities”. In our paper we outlined the gauge invariant approach to the construction of 
+the analog of guiding center operator in homogeneous/inhomogeneous magnetic fields. On the 
+face of it, the presented approach is unnecessary and superfluous, as e.g. there exists common 
+consensus that in the discussed “axial” problems (e.g., 𝑩(𝒓) = 𝐵0𝑥𝒛̂ 𝐿
+⁄ ) we must choose the wave 
+function in the form Ψ(𝒓) = 𝜑(𝑥)exp (𝑖𝜆𝑦) simply relying on symmetry considerations. We ar-
+gue that in accord with presented above considerations this is not so simple. Such naïve approach 
+is valid only for the “proper” chosen gauge. The phase dependence of the wave function factor is 
+determined by the eigenfunction of the additional time independent operator –pseudo-momentum, 
+which in its turn is explicitly dependent upon the particular choice of the vector potential form. 
+The steps to be taken to obtain the “proper” equations for the wave functions are as follows. First, 
+we are free to choose any form of the vector potential fulfilling condition 𝑟𝑜𝑡 𝑨(𝒓) = 𝑩(𝒓) ap-
+propriate to the considered magnetic field spatial distribution. Second, we define the conserved 
+pseudo-momentum operator (or its component) dependent upon chosen gauge but nevertheless by 
+the definition gauge invariant [57]. Fixing the phase of the exponential factor by imposing the 
+restriction that the sought wave functions are simultaneously the eigenfunctions of the pseudo-
+momentum, we arrive at last to the gauge invariant equation. It is easy to verify that following 
+these steps we always obtain the solutions which can be mapped in different gauges upon each 
+other by traditional Weyl gauge transformation. 
+ 
+
+13 
+ 
+Acknowledgements 
+The work has been supported in part by the Ministry of Science and Higher Education of the 
+Russian Federation under Project #FEUZ-2020-0054. 
+ 
+ 
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+

NNAyT4oBgHgl3EQf6_pQ/content/tmp_files/2301.00830v1.pdf.txt

@@ -0,0 +1,1274 @@
+Draft version January 4, 2023
+Typeset using LATEX twocolumn style in AASTeX631
+Evidence for AGN-Regulated Cooling in Clusters at z ∼ 1.4: A Multi-Wavelength View of
+SPT-CL J0607-4448
+Megan Masterson
+,1 Michael McDonald
+,1 Behzad Ansarinejad
+,2 Matthew Bayliss
+,3
+Bradford A. Benson
+,4, 5, 6 Lindsey E. Bleem
+,7, 6 Michael S. Calzadilla
+,1 Alastair C. Edge
+,8
+Benjamin Floyd
+,9 Keunho J. Kim
+,3 Gourav Khullar
+,10, 11, 1 and Taweewat Somboonpanyakul
+12
+1MIT Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
+2School of Physics, University of Melbourne, Parkville, VIC 3010, Australia
+3Department of Physics, University of Cincinnati, Cincinnati, OH 45221, USA
+4Fermi National Accelerator Laboratory, Batavia, IL 60510-0500, USA
+5Department of Astronomy and Astrophysics, University of Chicago, Chicago, IL 60637, USA
+6Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
+7High Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA
+8Department of Physics, Durham University, Durham DH1 3LE, UK
+9Department of Physics and Astronomy, University of Missouri–Kansas City, 5110 Rockhill Road, Kansas City, MO 64110, USA
+10Department of Astronomy and Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
+11Kavli Institute for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637
+12Kavli Institute for Particle Astrophysics & Cosmology, P.O. Box 2450, Stanford University, Stanford, CA 94305, USA
+ABSTRACT
+We present a multi-wavelength analysis of the galaxy cluster SPT-CL J0607-4448 (SPT0607), which is
+one of the most distant clusters discovered by the South Pole Telescope (SPT) at z = 1.4010 ± 0.0028.
+The high-redshift cluster shows clear signs of being relaxed with well-regulated feedback from the
+active galactic nucleus (AGN) in the brightest cluster galaxy (BCG). Using Chandra X-ray data, we
+construct thermodynamic profiles and determine the properties of the intracluster medium. The cool
+core nature of the cluster is supported by a centrally-peaked density profile and low central entropy
+(K0 = 18+11
+−9 keV cm2), which we estimate assuming an isothermal temperature profile due to the
+limited spectral information given the distance to the cluster. Using the density profile and gas cooling
+time inferred from the X-ray data, we find a mass cooling rate of
+˙Mcool = 100+90
+−60 M⊙ yr−1. From
+optical spectroscopy and photometry around the [O ii] emission line, we estimate that the BCG
+star formation rate is SFR[O II] = 1.7+1.0
+−0.6 M⊙ yr−1, roughly two orders of magnitude lower than
+the predicted mass cooling rate.
+In addition, using ATCA radio data at 2.1 GHz, we measure a
+radio jet power of Pcav = 3.2+2.1
+−1.3 × 1044 erg s−1, which is consistent with the X-ray cooling luminosity
+(Lcool = 1.9+0.2
+−0.5×1044 erg s−1 within rcool = 43 kpc). These findings suggest that SPT0607 is a relaxed,
+cool core cluster with AGN-regulated cooling at an epoch shortly after cluster formation, implying that
+the balance between cooling and feedback can be reached quickly. We discuss implications for these
+findings on the evolution of AGN feedback in galaxy clusters.
+Keywords: Brightest cluster galaxies (181)–Galaxy clusters (584)–Intracluster medium (858)–Radio
+galaxies (1343)–High-redshift galaxy clusters (2007)
+1. INTRODUCTION
+A galaxy cluster contains tens to hundreds of mem-
+ber galaxies (with some reaching over a thousand mem-
+Corresponding author: Megan Masterson
+[email protected]
+bers) surrounded by hot, ionized plasma called the in-
+tracluster medium (ICM), all embedded in a massive
+dark matter halo that constitutes the majority of the
+cluster mass. The ICM is the dominant baryonic com-
+ponent of clusters, and it is visible at X-ray wavelengths
+via bremsstrahlung radiation caused by the motion of
+charged particles. We often classify galaxy clusters into
+two main groups—cool core clusters, in which the cen-
+arXiv:2301.00830v1  [astro-ph.GA]  2 Jan 2023
+
+ID2
+Masterson et al.
+tral temperature drops and the density increases, and
+non-cool core clusters, which have cores that are roughly
+isothermal. In cool core clusters, the ICM has short ra-
+diative cooling times and should produce massive cool-
+ing flows of ˙M ∼ 100−1000 M⊙ yr−1, in which cold gas
+condenses out of the hot plasma (see Fabian 1994, for a
+review). However, such cooling flows are not observed
+in most systems, with typical star formation rates on
+the order of ∼ 1% the expected cooling rate (e.g. O’Dea
+et al. 2008; McDonald et al. 2018) and a lack of cool gas
+as probed with high resolution X-ray spectroscopy (e.g.
+Peterson et al. 2003; Bregman et al. 2006; Peterson &
+Fabian 2006).
+One of the dominant mechanisms that is thought to
+prevent the rapid cooling of the ICM is mechanical
+feedback from an active galactic nucleus (AGN) in the
+brightest cluster galaxy (BCG; e.g. McNamara & Nulsen
+2007, 2012; Fabian 2012). In this paradigm, the radio-
+loud AGN is accreting well below the Eddington limit
+and launches powerful jets that inject energy into the
+ICM by inflating bubbles and thus creating X-ray cavi-
+ties. Observationally, the inflation of these bubbles has
+been shown to have enough energy to balance the cool-
+ing flow in many systems (e.g. Bˆırzan et al. 2004; Dunn
+& Fabian 2006; Rafferty et al. 2006; Hlavacek-Larrondo
+et al. 2012, 2015). Although AGN feedback is now gen-
+erally accepted as one of the main heating mechanisms
+balancing cooling in clusters of galaxies, there are still
+many open questions, including how the properties of
+the ICM and the impact of AGN feedback have evolved
+over cosmic time.
+The study of high-redshift galaxy clusters and cluster
+evolution has been greatly aided by recent advances in
+the millimeter/sub-millimeter regime, whereby the ther-
+mal Sunyaev-Zel’dovich (SZ) effect can be used to detect
+galaxy clusters using their imprint on the cosmic mi-
+crowave background (Sunyaev & Zeldovich 1972). Mil-
+limeter observatories like the Planck satellite (Planck
+Collaboration et al. 2016), the Atacama Cosmology
+Telescope (ACT; Hilton et al. 2018, 2021), and the South
+Pole Telescope (SPT; Carlstrom et al. 2011; Bleem et al.
+2015, 2020; Huang et al. 2020) have greatly increased
+the number of detected galaxy clusters at z > 1. The
+SZ selection method is mass-limited, nearly redshift-
+independent (e.g. Bleem et al. 2015), and independent
+of the dynamical state of the cluster (e.g. Nurgaliev
+et al. 2017), allowing for a selection function well-suited
+for cluster evolution studies. In addition, SZ detection
+avoids significant bias toward strong cool core systems
+(e.g. Lin et al. 2015), which plagues X-ray detection
+mechanisms (e.g. Eckert et al. 2011), and avoids any
+bias due to cluster galaxy properties that are present in
+optical and infrared detection methods.
+Uniform X-ray follow-up of SZ-selected clusters has
+revealed similarity among ICM thermodynamic prop-
+erties and the impact of AGN feedback on the ICM
+from z ∼ 0 up to z ∼ 1.7 (e.g. McDonald et al. 2013;
+Hlavacek-Larrondo et al. 2015; McDonald et al. 2017;
+Ruppin et al. 2021; Ghirardini et al. 2021). In partic-
+ular, the density profiles of clusters are consistent with
+self-similar evolution in the outskirts and with no red-
+shift evolution in the cores (McDonald et al. 2017; Rup-
+pin et al. 2021), indicating consistent non-gravitational
+processes at play in cluster cores responsible for the devi-
+ation from self-similarity. Likewise, Hlavacek-Larrondo
+et al. (2015) found that the power from AGN feedback
+in cool core clusters has been roughly constant up to
+z ∼ 1. Probing the ICM in the most distant clusters
+will be a primary focus of next generation X-ray missions
+like Athena (Barret et al. 2020). For now, focusing on
+multi-wavelength observations of the most distant clus-
+ters allows us to place constraints on the nature of AGN
+feedback and ICM properties at z > 1.
+SPT-CL J0607-4448 (hereafter SPT0607) is one of the
+most distant SPT clusters discovered to date (Bleem
+et al. 2015), with a redshift of z = 1.4010 ± 0.0028
+as measured by spectroscopic follow-up of cluster mem-
+bers (Khullar et al. 2019). As such, it has been exten-
+sively followed up with various observatories and has
+been studied in the X-ray as part of the SPT-SZ high-z
+sample (McDonald et al. 2017; Ghirardini et al. 2021).
+In the optical band, SPT0607 seems to contain two main
+groups of galaxies, one at z = 1.401 and one closer to
+z ∼ 1.48. However, the red sequence, dynamics of the
+cluster members, and spectroscopy of the BCG favors
+the lower redshift solution (Khullar et al. 2019; Straz-
+zullo et al. 2019). Finally, the galactic properties of clus-
+ter members were investigated in Strazzullo et al. (2019),
+where they found an overdensity of red galaxies in the
+cluster, although this overdensity was less prominent
+than other clusters in their sample (with 1.4 ≲ z ≲ 1.7)
+despite SPT0607 having the most massive BCG. Our
+analysis of SPT0607 brings together multi-wavelength
+observations to put together the full picture of this re-
+laxed, cool core cluster with well-regulated cooling and
+feedback at such a high redshift.
+This work is organized as follows. In Section 2, we
+outline the multi-wavelength data analyzed in this work.
+We present our results in Section 3 and discuss the impli-
+cations of these findings on our understanding of cluster
+evolution and the AGN feedback process at high red-
+shift in Section 4. Finally, we summarize our findings
+in Section 5. Throughout this work, we utilize a ΛCDM
+
+A Multi-Wavelength View of SPT-CL J0607-4448
+3
+cosmology with H0 = 70 km s−1 Mpc−1, ΩM = 0.3, and
+ΩΛ = 0.7. All quoted uncertainties correspond to 68%
+(1σ) confidence, unless otherwise noted.
+2. OBSERVATIONS & DATA REDUCTION
+In Figure 1, we show the X-ray, optical/infrared (IR),
+and radio data used in this analysis of SPT0607. On
+the left and right, we show the Chandra X-ray data
+and ATCA radio data, respectively, and locate the as-
+sociated peaks in green (X-ray) and magenta (radio).
+The center panel shows an RGB image using 3 HST
+filters (F140W, F110W, and F814W), with the same lo-
+cations of the X-ray and radio peaks overplotted. Both
+the X-ray and radio peak are coincident with the BCG
+of SPT0607, as expected for a well-regulated cool core
+cluster. In the rest of this section, we describe the data
+and reduction methods used in this paper.
+2.1. Chandra X-ray Observations
+SPT0607 was observed with the Chandra ACIS-I in-
+strument for a total of 112.5 ksec in January and Febru-
+ary 2016. The details of the observations used in this
+analysis are provided in Table 1. We reduced and ana-
+lyzed these data using CIAO (version 4.12; Fruscione
+et al. 2006) and calibration files from CALDB (ver-
+sion 4.9.2.1). All observations were taken in VFAINT
+mode so we applied additional improved background fil-
+tering. We detected and removed point sources using
+the wavdetect tool and sigma-clipped the light curve
+at 3σ with the lc clean tool to remove any periods of
+background flaring from our good time intervals (GTIs).
+At z = 1.401 (Khullar et al. 2019), the angular ex-
+tent of the cluster is relatively small compared to the
+ACIS-I array, taking up only a single detector chip.
+Thus, we used an off-source region on the remaining
+other 3 detectors to produce the background spectra.
+We extracted source and background X-ray spectra in
+the 0.5-7.0 keV energy range and used XSPEC (ver-
+sion 12.11.1) for spectral fitting. Spectra were grouped
+to a minimum of 1 count per bin and C-statistic min-
+imization was used for fitting (Cash 1979).
+We used
+the XSPEC model phabs(apec), where the phabs com-
+ponent accounts for absorption in the Milky Way and
+the apec model accounts for the emission from the in-
+tracluster medium. Abundances were taken from An-
+ders & Grevesse (1989). The absorption column density
+for the phabs model was free to vary between galac-
+tic NHI value, NH = 6.78 × 1020 cm−2 (HI4PI Col-
+laboration et al. 2016), and the galactic NH, tot value,
+NH, tot = NHI + NH2 = 8.33 × 1020 cm−2 (Willingale
+et al. 2013). For the cluster emission, we fixed the red-
+Table 1. Chandra Observation Information
+ObsID
+Date
+Cleaned Exposure Time
+(ksec)
+17210
+2016-02-04
+37.4
+17499
+2016-01-30
+39.3
+17500
+2016-02-20
+17.8
+18770
+2016-02-22
+18.0
+shift to z = 1.401 and the metallicity to Z = 0.3Z⊙
+given the limited data quality.
+2.2. Optical and Infrared Photometry
+SPT0607 was observed with the Hubble Space Tele-
+scope (HST) in four different broad-band filters with
+Proposal IDs 14252 (PI: V. Strazzullo) and 14677 (PI:
+T. Schrabback). The cluster was observed in the opti-
+cal to near-infrared (rest-frame) with the F606W and
+F814W filters using the Advanced Camera for Surveys
+(ACS) and with the F110W and F140W filters using
+the Wide Field Camera 3 (WFC3). The data were re-
+duced using the AstroDrizzle package to remove cos-
+mic rays, perform standard data reduction, and com-
+bine images. We utilize the HST photometry primarily
+to understand the optical spectral energy distribution
+(SED) of the BCG and calibrate our ground-based spec-
+troscopy.
+The BCG of SPT0607 is undetected in the
+bluest filter, F606W, leading to a 1σ upper limit on the
+flux of Fλ, F606W > 9.1 × 10−20 erg s−1 cm−2 ˚A−1.
+2.3. Optical Spectroscopy
+Optical spectra of potential cluster members of
+SPT0607 were obtained using the Low Dispersion Sur-
+vey Spectrograph (LDSS-3C) on the 6.5m Magellan
+Clay Telescope (Khullar et al. 2019).
+The VPH-Red
+grism was used, providing nominal wavelength cover-
+age from 6,000 – 10,000 ˚A. With SPT0607 at a red-
+shift of z = 1.401, this wavelength coverage provides
+access to the [O ii] emission line, which was used to es-
+timate the amount of star formation in the BCG. How-
+ever, these spectra, initially designed for cluster confir-
+mation by measuring the redshift of potential cluster
+members, were only wavelength-calibrated and not flux-
+calibrated. Therefore, in order to obtain a line flux for
+[O ii] to estimate star formation rates, we utilized the
+HST photometry to roughly calibrate the spectrum flux.
+We first measured an equivalent width from the uncali-
+brated LDSS-3C spectrum, and then fit the three-band
+HST photometry to a SED with an old and young stel-
+
+4
+Masterson et al.
+0.5-7.0 keV
+Chandra
+500 kpc (~1’)
+F814W, F110W, F140W
+HST
+100 kpc (~12")
+1-3 GHz
+ATCA
+500 kpc (~1’)
+Figure 1. Left: Merged Chandra X-ray counts image in the broad-band 0.5-7.0 keV. The image is binned such that each pixel
+is 0.′′984 on each side and then smoothed with a Gaussian kernel of 4 pixels. The green “×” shows the location of the X-ray
+peak, which we use as the center for all X-ray profiles. Middle: RGB image of SPT0607 using the HST F140W (red), F110W
+(green), F814W (blue) filters. The green “×” shows the location of the X-ray peak and the magenta “+” shows the location of
+the radio peak, both of which are coincident with the BCG of SPT0607. Right: ATCA 2 GHz radio image with the synthesized
+beam in orange in the lower right corner. The magenta “+” shows the location of the radio peak.
+lar population (10 Gyr and 10 Myr, respectively) de-
+rived from the Starburst99 models (Leitherer et al.
+1999). As the BCG in SPT0607 was undetected in the
+F606W filter, we used only the F814W, F110W, and
+F140W photometry measurements from HST to fit the
+SED, which was constrained to within roughly 10% at
+the 1σ level around the rest-frame wavelength of [O ii]
+(see Figure 5 and Section 3.3). This provided a mea-
+sure of the expected continuum flux at the wavelength
+of [O ii], which thus allowed us to convert the equiva-
+lent width of the [O ii] emission line in the LDSS-3C
+spectrum to a line flux.
+2.4. Radio Observations
+SPT0607 was observed with the Australia Telescope
+Compact Array (ATCA) in the 6A configuration in the
+1–3 GHz band on 20th August 2016 in seven 20 min
+visits spread evenly over an 8.5 hour period. These data
+provide a beam of 6′′ × 3.′′5 at 2 GHz. The data were
+reduced with the 05/21/2015 release of the Miriad soft-
+ware package (Sault et al. 1995).
+The phase calibra-
+tor 0647-475 was used to create the radio maps, with
+some multi-faceting, but no self-calibration was neces-
+sary. The rms value for the resulting image is 23 µJy
+with a dynamic range of ∼3000, ensuring sensitivity to
+extended emission.
+3. RESULTS
+3.1. ICM Properties & Thermodynamic Profiles
+In this section, we present the results of the X-ray
+data analysis whereby we measure the properties of the
+ICM in SPT0607. We are focused on the core properties
+of SPT0607, where the impact of AGN feedback is most
+prevalent, and hence, we measured our radial profiles
+with respect to the X-ray peak location, as marked in
+the left and middle panels of Figure 1.
+As has been
+noted previously (e.g. McDonald et al. 2013; Sanders
+et al. 2018; Ruppin et al. 2021), using a center based on
+the large scale X-ray centroid, as was done in McDonald
+et al. (2017) and Ghirardini et al. (2021), gives a slightly
+different profile and leads to lower central density and
+higher central entropy. Additionally, we note that given
+the relatively high number of counts from SPT0607 (∼
+700), our peak location is robust to variations due to
+noise (e.g. Ruppin et al. 2021).
+Due to the high redshift of the source, we make a few
+conservative assumptions with respect to the temper-
+ature profile of the cluster. We first assume that the
+temperature profile is isothermal, where the tempera-
+ture is a core-excised temperature measured within a
+radius (0.15 − 1)R500, using R500 = 0.56 Mpc from Mc-
+Donald et al. (2017). Although this is likely a poor as-
+sumption for the true nature of the temperature profile
+in SPT0607, it provides a strong upper bound on many
+of our measured thermodynamic properties. In reality,
+we believe that the cluster has a strong cool core due
+to the excess surface brightness, radio jet, and lack of
+significant star formation features in the BCG. We then
+show in the remainder of this section that we can still
+recover the features of a strong cool core even with this
+assumption of an isothermal temperature profile, pro-
+viding compelling evidence for the cool core nature of
+this system. After showing that SPT0607 does indeed
+host a cool core, we also assume a standard cool core
+
+A Multi-Wavelength View of SPT-CL J0607-4448
+5
+temperature profile (Vikhlinin et al. 2006), scaled to the
+global, core-excised temperature, to obtain a better es-
+timate of the central thermodynamic properties.
+3.1.1. Global Temperature Measurement
+As detailed in Section 2.1, we fit the cluster X-ray
+spectrum in the core-excised region with the simple
+model phabs(apec) for cluster emission, with the red-
+shift fixed at z = 1.401.
+Cluster metallicity is typi-
+cally constrained by the highly ionized Fe K-shell lines
+in X-ray spectra of the ICM, but is poorly constrained
+in our fits given the high redshift of SPT0607. Thus,
+we fixed the metallicity at Z = 0.3Z⊙, motivated by
+detailed low redshift studies, which find that the aver-
+age cluster metallicity is roughly a third of the solar
+value (e.g. Mushotzky & Loewenstein 1997; De Grandi
+& Molendi 2001; Urban et al. 2017), and recent metal-
+licity evolution studies, which show little evolution in
+the cluster metallicity out to z ∼ 1 (e.g. McDonald
+et al. 2016a; Flores et al. 2021).
+The ICM metal-
+licity has been shown to have a weak dependence on
+temperature (e.g. Fukazawa et al. 1998), and hence,
+this choice likely has little impact on our measured
+global temperature.
+Following the methodology out-
+lined in Section 2.1, we find a core-excised temperature
+of ⟨kT⟩ = 6.75+2.14
+−1.51 keV. Using the higher redshift value
+for SPT0607 of z = 1.48 for the cluster redshift (see Sec-
+tion 1), we measure a slightly higher core-excised tem-
+perature of ⟨kT⟩ = 8.07+6.30
+−2.76 keV, but this is consistent
+with our initial estimate within 1σ uncertainty. Using
+both Chandra and XMM-Newton data, Ghirardini et al.
+(2021) found a temperature of T0 = 6.0 ± 0.8 keV when
+fitting a Vikhlinin cool core temperature profile, which
+is consistent with our measurement when considering
+the differences in the temperature estimates (Vikhlinin
+et al. 2006).
+3.1.2. Emission Measure and Density Profiles
+To derive an emission measure from the X-ray data,
+we extracted a spectrum from each observation in radial
+bins. We used extraction bins with outer radii defined
+by
+rout,i = (a + bi + ci2 + di3)R500
+(1)
+where the constants a, b, c, and d are as defined in Mc-
+Donald et al. (2017), R500 = 560 kpc (McDonald et al.
+2017), and i = 1, 2, ..., 17. We use fewer radial annuli
+than in McDonald et al. (2017) due to poor signal-to-
+noise in the cluster outskirts for SPT0607. In each radial
+bin, we fit the spectrum for all 4 observations simulta-
+neously, with all parameters tied across all observations.
+To derive an emission measure, we simply fix the tem-
+perature to the global, core-excised temperature previ-
+ously described and fit only to the normalization of the
+apec model. The normalization of the apec model has
+astrophysical meaning and is given by
+norm =
+10−14
+4π [DA (1 + z)]2
+�
+nenHdV,
+(2)
+where DA is the angular distance to the source in units
+of cm, ne is the electron density in cm−3, and nH is the
+H density in cm−3. Then, by assuming a spherical ge-
+ometry, the normalization can be related to the emission
+measure, which is given by
+EM =
+�
+nenHdl,
+(3)
+where the integral here is along the line of the sight
+through the cluster. Thus, we can use the apec nor-
+malization to obtain the emission measure for each ra-
+dial bin. Because the normalization measurement is de-
+pendent on the temperature we use, we also account
+for the uncertainty in the temperature measurement by
+including an additional 10% uncertainty on each apec
+normalization measurement (the average difference be-
+tween the normalization at ⟨kT⟩ and the normalization
+at ⟨kT⟩ ± 1σ for the isothermal temperature).
+To fit the emission measure, we use the modified β-
+model (Vikhlinin et al. 2006), whereby the density is
+given by
+nenH = n2
+0
+(r/rc)−α
+(1 + r2/r2c)3β−α/2
+1
+(1 + r3/r3s)ϵ/3 ,
+(4)
+where n0 is the central density, rc and rs are scaling
+radii for the cluster core and outskirts, and r is the ra-
+dial coordinate. This model for the density is then pro-
+jected and integrated numerically along the line of sight
+to create an emission measure model.
+We utilize the
+Markov Chain Monte Carlo (MCMC) implementation
+emcee from Foreman-Mackey et al. (2013) to perform
+the fitting.
+We use uniform priors on all parameters
+and a Gaussian likelihood, given by
+L = −1
+2χ2 = −1
+2
+N
+�
+i=1
+�EMmeasured − EMmodel
+σEM
+�2
+,
+(5)
+where σEM are our errors on the emission measure.
+We first maximize this likelihood function for our data
+and then use the maximum likelihood parameters with
+some scatter as our initial position for the walkers in the
+MCMC chain. We run the chain with 32 walkers, each
+for 5 × 105 chain steps after a burn length of 5 × 104
+chain steps (which is significantly longer than the inte-
+grated autocorrelation time of the resulting chain). The
+resulting fit to the emission measure is shown in the left
+panel of Figure 2.
+
+6
+Masterson et al.
+100
+101
+102
+103
+Radius (kpc)
+1017
+1018
+1019
+1020
+1021
+1022
+Emission Measure (cm−5)
+68% confidence interval
+95% confidence interval
+Median fit
+Maximum likelihood
+100
+101
+102
+103
+Radius (kpc)
+10−4
+10−3
+10−2
+10−1
+100
+ne (cm−3)
+68% confidence interval
+95% confidence interval
+Median fit
+Maximum likelihood
+Ghirardini et al. (2021)
+Figure 2. Left: The emission measure fit for SPT0607. The emission measure is computed by using the APEC normalization
+in each of the imaging bins and fitting a projected density profile by integrating along the line of sight through the cluster. The
+red dashed line shows the maximum likelihood fit, using the Gaussian likelihood given in Equation 5. The profile with median
+fit parameters from the MCMC fit is shown in black, and the confidence interval from the MCMC chain at each radius for
+68% and 95% confidence is shown in the shaded regions. Right: The density profile for SPT0607, computed from the emission
+measure fit. The maximum likelihood profile is again shown in red, the median MCMC profile is shown in black, and the 68%
+and 95% confidence intervals are shown in the shaded regions. The comparison to the density profile from Ghirardini et al.
+(2021) is shown in blue. The discrepancy between the two profiles in the core is likely due to our different choice of center (see
+Section 3.1.2)
+We can easily turn our emission measure fit into a gas
+density profile for the cluster since we have fit parame-
+ters directly related to the density via Equation 4. For
+an ionized plasma with a metallicity of 0.3Z⊙, ne and
+nH are related via ne = ZnH, where Z = 1.199 is the
+average nuclear mass. Likewise, the total gas density of
+the system can be described by ρg = mpneA/Z, where
+mp is the mass of a proton and A = 1.397 is the av-
+erage nuclear charge. Our density profile is shown in
+the right panel of Figure 2, with a comparison to the
+density profile from Ghirardini et al. (2021), which uti-
+lizes both Chandra and XMM-Newton data. Ghirardini
+et al. (2021) use a large-scale centroid to compute their
+radial profiles, whereas we choose an X-ray peak ap-
+proach to capture the core properties. We find decent
+agreement at the majority of the cluster radii, although
+our profile predicts a larger overdensity in the cluster
+core. When using a centroid-based approach (i.e. the
+Ghirardini et al. (2021) center), we find better agree-
+ment between the two profiles, suggesting that the dis-
+crepancy in Figure 2 is due to our choice of using the
+X-ray peak as the cluster center rather than the large-
+scale centroid.
+3.1.3. Entropy Profile
+With the density profile for the cluster, we derive an
+entropy profile, which can both give us insight into the
+cool core nature of the cluster and trace the thermo-
+dynamic history of the ICM (Cavagnolo et al. 2009).
+Cluster entropy is defined as
+K = kT
+n2/3
+e
+.
+(6)
+Assuming an isothermal temperature profile provides an
+upper limit on the true entropy profile in the core of the
+cluster. Figure 3 shows the entropy profile for SPT0607
+using the isothermal temperature profile described in
+Section 3.1.1 and discretizing the entropy in the same
+bins as we used to measure the emission measure. We
+find good agreement in the cluster outskirts with the
+self-similar K ∝ R1.1 expectation (Voit et al. 2005). In
+the center, we find slight excess entropy compared to
+the self-similar expectation, with a central entropy of
+K0 = 18+11
+−9
+keV cm2 in the smallest bin (r ≈ 10 kpc).
+Thus, even with the most conservative assumption of
+an isothermal temperature profile, we still recover a low
+entropy core, consistent with the central entropy in the
+strong cool cores in the sample from Hudson et al. (2010)
+(K0 ≲ 22 keV cm2). This indicates that SPT0607 is
+indeed a strong cool core cluster.
+
+A Multi-Wavelength View of SPT-CL J0607-4448
+7
+100
+101
+102
+103
+Radius (kpc)
+100
+101
+102
+103
+K (keV cm2)
+68% confidence interval
+95% confidence interval
+Median fit
+Maximum likelihood
+K ∝ R1.1 (Voit+05)
+Figure 3. The entropy profile for SPT0607, computed from
+the derived density profile and an isothermal temperature
+profile. The analytic profile has been discretized in the same
+binning scheme used to fit the emission measure data. We
+find a low entropy core and good agreement in the cluster
+outskirts with the expected K ∝ R1.1 relation from Voit
+et al. (2005).
+To obtain a more accurate estimate of the central en-
+tropy, we also computed the entropy profile assuming
+that the temperature followed the Vikhlinin et al. (2006)
+cool core profile. Under this assumption, we find a cen-
+tral entropy K0 = 10+3
+−6 keV cm−2, which is again con-
+sistent with a strong cool core in SPT0607.
+3.1.4. Cooling Time
+The last key thermodynamic quantity that we com-
+pute is the cooling time, which is used to estimate rcool
+so that we can measure a mass cooling rate to compare
+with other indicators of cooling to get an idea of the
+suppression caused by AGN feedback. We compute the
+cooling time for the cluster using
+tcool = 3
+2
+(ne + nH)kT
+nenHΛ(T, Z),
+(7)
+where Λ(T, Z) is the cooling function for an astrophysi-
+cal plasma at a temperature T and metallicity Z, which
+we tabulate from Sutherland & Dopita (1993) for the
+closest temperature and metallicity for SPT0607. The
+cooling time profile we derive with an isothermal tem-
+perature profile is shown in Figure 4.
+Using this cooling time profile, we measure a cooling
+radius of rcool = 43+17
+−11 kpc, which is defined as the ra-
+dius at which the cooling time is equal to 3 Gyr.
+A
+100
+101
+102
+103
+Radius (kpc)
+10−1
+100
+101
+102
+tcool (Gyr)
+68% confidence interval
+95% confidence interval
+Median fit
+Maximum likelihood
+tcool = 3 Gyr
+rcool
+Figure 4. The cooling time profile for SPT0607, computed
+assuming an isothermal temperature profile and density pro-
+files derived in Section 3.1.2. The radius corresponding to
+tcool = 3 Gyr is shown with a blacked dotted line, with the
+corresponding 68% confidence interval shown in grey.
+cooling time of 3 Gyr was chosen as it has been shown
+to contain the most extended tracers of thermal insta-
+bilities in the ICM (e.g. McDonald et al. 2010, 2011).
+To obtain a mass cooling rate, we then integrate the gas
+density profile to within the cooling radius and compute
+the mass cooling rate using
+˙Mcool = Mgas(r < rcool)
+3 Gyr
+.
+(8)
+From this, we estimate from the X-ray analysis that
+the expected mass cooling rate is
+˙Mcool = 100+90
+−60 M⊙
+yr−1. Similarly to the central entropy, we also compute
+this value using a scaled version of the universal cool
+core temperature profile and find consistent mass cool-
+ing rates under that assumption.
+3.2. Radio Power
+We utilize ATCA 2.1 GHz observations of SPT0607
+to determine the total radio power associated with the
+BCG in SPT0607. The jet from the BCG is unresolved,
+and we measure an integrated flux using CASA (Mc-
+Mullin et al. 2007) of S2.1 GHz = 0.23 ± 0.11 mJy within
+an ovular aperture equal to the beam size centered on
+the radio peak. This corresponds to a 2.1 GHz radio
+luminosity of L2.1 GHz = (2.3 ± 1.1) × 1024 W Hz−1. We
+then estimate the radio power using
+Pν0 = 4πD2
+L(1 + z)α−1Sν0ν0,
+(9)
+
+8
+Masterson et al.
+from Cavagnolo et al. (2010), where ν0 is the observed
+frequency (2.1 GHz), Sν0 is the flux density at the ob-
+served frequency, DL is the luminosity distance, and α
+is the spectral index. Since we only have data at one
+frequency from ATCA, we cannot measure the spectral
+index, but instead adopt a typical value for extragalactic
+radio galaxies of α = 0.8 as in Cavagnolo et al. (2010).
+Using a spectral index of α = 0.8, we find a radio power
+of P2.1 GHz = (4.8 ± 2.4) × 1040 erg s−1.
+To compare the power of the radio jet in the BCG to
+the amount of cooling expected in the ICM, we use the
+scaling relation from Cavagnolo et al. (2010) to convert
+the measured radio power to a jet power. We first use
+the same spectral index to convert the observed 2.1 GHz
+power to a 1.4 GHz power, which can then be directly
+converted to jet power using Equation (1) of Cavagnolo
+et al. (2010) given by
+log Pcav = (0.75 ± 0.14) log P1.4 + (1.91 ± 0.18),
+(10)
+where Pcav is in units of 1042 erg s−1 and P1.4 is in
+units of 1040 erg s−1. We find a jet power of Pcav =
+3.2+2.1
+−1.3 × 1044 erg s−1 using this scaling relation.
+To
+compare the heating from the radio jet to the cooling
+of the ICM, we compute the X-ray cooling luminosity
+of the ICM within rcool, using our derived value of rcool
+from Section 3.1.4. We find an unabsorbed X-ray cooling
+luminosity of Lcool = 1.9+0.2
+−0.5 × 1044 erg s−1 in the 0.01-
+100 keV band, which is identical to the radio jet power
+within 1σ confidence. This is consistent with the radio
+BCG power versus X-ray cooling luminosity found in
+a large sample of low redshift clusters in Hogan et al.
+(2015), as well as with the lack of a significant redshift
+evolution in Pcav/Lcool for clusters out to z ∼ 1.3 in
+Ruppin et al. (2022). The implications of these findings
+on the regulation of cooling in SPT0607 by radio-mode
+AGN feedback are discussed further in Section 4.
+3.3. Regulated Star Formation in the BCG
+Using the LDSS-3C optical spectrum from the Magel-
+lan Clay telescope, we estimate the star formation rate
+(SFR) in the BCG by measuring a luminosity of the
+[O ii] λλ3727, 3729 ˚A doublet. The [O ii] emission fea-
+ture is a useful indicator of star formation (e.g. Kenni-
+cutt 1998; Kewley et al. 2004), especially in the high-
+redshift universe because it has a similar ionization en-
+ergy to hydrogen, but, unlike the Hα transition, is not
+redshifted out of the optical band. The [O ii] emission
+traces warm gas with T ∼ 104 K around young O and
+B stars, thus tracing instantaneous star formation on
+timescales on the order of ∼ 10 Myr. However, SFRs
+derived from [O ii] emission line are more dependent on
+dust, metallicity, and ionization than other tracers like
+Hα, UV, and far-IR luminosities (e.g. Rosa-Gonz´alez
+et al. 2002; Kewley et al. 2004; Moustakas et al. 2006),
+which we cannot accurately determine with current data
+on SPT0607. AGN can also excite [O ii] in the nuclei of
+galaxies, but the AGN in SPT0607 is radiatively ineffi-
+cient and weak in X-ray emission. Thus, we do not ex-
+pect the central AGN to be contributing significantly to
+the [O ii] emission in SPT0607 and can safely attribute
+the majority of the [O ii] emission to star formation.
+We fit the LDSS-3C spectrum within 100 ˚A on ei-
+ther side of the expected [O ii] emission feature with a
+constant to estimate the continuum and doublet Gaus-
+sian feature for the [O ii] line. We fix the redshift at
+z = 1.401 for the cluster, and allowed the line centers to
+vary within 500 km s−1 of the atomic value to account
+for peculiar motions in the cluster. We restrict the width
+of the line to be less than 500 km s−1 to account for tur-
+bulent motions broadening the line. We tie the widths
+of the two Gaussian components together and allowed
+their line ratio to be free. We use the emcee package
+(Foreman-Mackey et al. 2013) with a Gaussian likeli-
+hood and uniform, uninformative priors to fit the spec-
+trum using an MCMC approach with 32 walkers, 50,000
+chain steps per walker, and a burn length of 5,000 chain
+steps per walker (which is significantly longer than the
+integrated autocorrelation time of the resulting chain).
+The result of the fit is shown in the top panel of Fig-
+ure 5. We detect a relatively weak emission feature in
+[O ii] with a velocity offset of v = −200 ± 60 km s−1, a
+line width of 230 ± 40 km s−1, and a rest-frame equiva-
+lent width (EW) of EW[OII] = 6.0 ± 0.9 ˚A. This equiv-
+alent width is then turned into a line flux using the
+flux-calibrated HST photometry to model the contin-
+uum SED, as shown in the bottom panel of Figure 5
+and detailed in Section 2.3.
+From this calibration, we measure an [O ii] luminos-
+ity of L[OII] = 1.3+0.3
+−0.2 × 1041 erg s−1, which has not
+been corrected for extinction. We account for extinc-
+tion by folding in uncertainty on E(B −V ) by assuming
+a uniform distribution between E(B−V ) = 0 (i.e. dust-
+free) and E(B − V ) = 0.3. Using Equations (10) and
+(17) of Kewley et al. (2004), we convert our observed
+[O ii] luminosity to a SFR (assuming a solar value of
+log(O/H) + 12 = 8.9). From our MCMC chains from
+fitting the line and folding in the uniform distribution of
+E(B−V ), we obtain an extinction-corrected star forma-
+tion rate of SFR[O ii] = 1.7+1.0
+−0.6 M⊙ yr−1. This value is
+more than two orders of magnitude lower than the cool-
+ing rate we measure in the X-ray band, indicating that
+the cooling in SPT0607 is well-regulated by AGN feed-
+back. Likewise, this star formation rate is comparable
+to low-redshift samples of BCGs with little on-going star
+
+A Multi-Wavelength View of SPT-CL J0607-4448
+9
+8875
+8900
+8925
+8950
+8975
+9000
+9025
+9050
+Observed Wavelength (˚A)
+0
+500
+1000
+1500
+2000
+2500
+Flux (du/pixel)
+[OII] Emission at z = 1.401
+Maximum Likelihood
+68% Confidence Interval
+95% Confidence Interval
+Data
+6000
+8000
+10000
+12000
+14000
+Observed Wavelength (˚A)
+1039
+1040
+1041
+Lλ (erg s−1 ˚A−1)
+Old Population
+Young Population
+Total Model
+68% Confidence Interval
+95% Confidence Interval
+HST Data
+Figure 5. Top: Fit to the wavelength-calibrated LDSS-3C
+spectrum around the [O ii] emission feature. The maximum
+likelihood fit is shown as a red dashed line, with the two in-
+dividual Gaussian components shown with red dotted lines.
+Confidence intervals are shown in green. The observed wave-
+length of the [O ii] doublet is shown with blue dotted lines.
+We allow for some systematic offset from the observed wave-
+length to account for motion within the cluster.
+Bottom:
+Fit to the three-band HST photometry using a simple young
+and old stellar population model from Starburst99 (Lei-
+therer et al. 1999). The total model is shown in black, with
+confidence intervals in green. The young and old stellar pop-
+ulation contributions are shown in blue and red, respectively.
+A 1σ upper bound from the F606W filter is also shown, al-
+though this is not used in the fitting procedure. The SED
+fit is used to obtain a continuum flux at the wavelength of
+[O ii], with which we can combine the equivalent width mea-
+surement from the top panel to determine the [O ii] line flux.
+formation as measured with Hα and other SFR indica-
+tors (e.g. Crawford et al. 1999; McDonald et al. 2010).
+This thus adds to the evidence that SPT0607 is a high-
+redshift analog of the large population of relaxed, low-
+redshift clusters with well-regulated star formation and
+ICM cooling by AGN feedback.
+4. DISCUSSION
+From the analysis of X-ray, optical, and radio obser-
+vations, SPT0607 clearly hosts a strong cool core with
+AGN feedback offsetting the cooling from the ICM, as
+is common place in low redshift galaxy clusters.
+An
+overview of properties of the cluster and BCG derived
+in this work are given in Table 2, highlighting the low
+central entropy, similarity of the radio cavity power and
+cooling luminosity, and the SFR that is ∼1% of the pre-
+dicted mass cooling rate. In the remainder of this sec-
+Table 2. Summary of Cluster and BCG Properties
+BCG Property
+Value
+Central Entropy
+K0 = 18+11
+−9 keV cm2
+X-ray Mass Cooling Rate
+˙Mcool = 100+90
+−60 M⊙ yr−1
+X-ray Cooling Luminosity
+Lcool = 1.9+0.2
+−0.5 × 1044 erg s−1
+Radio Jet Power
+Pcav = 3.2+2.1
+−1.3 × 1044 erg s−1
+Star Formation Rate
+1.7+1.0
+−0.6 M⊙ yr−1
+tion, we discuss the implications that these findings have
+on our understanding of high redshift clusters and the
+evolution of AGN feedback.
+4.1. Constraints on the Onset of Radio-Mode Feedback
+At low redshifts, radio-mode AGN feedback, whereby
+the central AGN accretes mass at a low rate and
+launches radio jets that deposit large amounts of me-
+chanical energy into the ICM, is the main mechanism
+by which runaway ICM cooling is prevented in cool core
+clusters (e.g. Bˆırzan et al. 2004; Dunn & Fabian 2006;
+Rafferty et al. 2006). Through multi-wavelength obser-
+vations, we have shown that SPT0607 has well-regulated
+radio-mode feedback from its BCG and, to our knowl-
+edge, is the highest redshift cluster with these properties
+known to date. As such, it provides one of the strongest
+constraints to date on the onset of AGN feedback in
+galaxy clusters.
+Simulations and theoretical models of the evolution
+of AGN feedback and supermassive black hole growth
+suggest that on average AGN in cluster environments
+should transition from quasar-mode feedback at early
+times, where the black hole is accreting at higher rates
+and the accretion process is radiatively efficient, to
+radio-mode feedback at late times (e.g. Churazov et al.
+2005; Croton et al. 2006). Recent simulations suggest
+that this transition should take on the order of 1-2 Gyr
+to occur in BCGs in cool core clusters (e.g. Qiu et al.
+2019). Indeed, at low redshifts, only on the order of 1-2%
+of clusters are observed to have a X-ray bright central
+AGN, which is expected for radiatively efficient accre-
+tion in the BCG and quasar-mode feedback (e.g. Green
+et al. 2017; Somboonpanyakul et al. 2021). SPT0607 has
+well-regulated radio-mode feedback from its BCG, sug-
+gesting that the radio-mode feedback must be present
+and a dominant form of AGN feedback in some clusters
+out to at least z = 1.4. Whether this is the dominant
+mechanism of feedback in most high redshift systems
+is a question that still remains to be answered with a
+more complete sample of radio and X-ray observations
+
+10
+Masterson et al.
+of high redshift clusters. However, we can use SPT0607
+to place constraints on the minimum redshift at which
+AGN feedback must have turned on in clusters; under
+the assumption that BCGs are dominated by radiatively
+efficient accretion during the first 1-2 Gyrs (Qiu et al.
+2019), the lowest redshifts at which the AGN feedback
+process could have began in SPT0607 is z ∼ 1.9 − 2.6.
+Previously, studies of X-ray cavities from jet-powered
+bubbles in the ICM have shown there is little evolution
+in the properties of radio-mode feedback from the local
+universe back to z ∼ 0.8 (Hlavacek-Larrondo et al. 2012,
+2015). Additionally, the discovery of more distant cool
+core clusters with central radio sources capable of bal-
+ancing ICM cooling, such as WARPJ1415.1+3612, have
+extended these findings out to z ∼ 1 (Santos et al. 2012).
+With SPT0607, we can extend this relation even further
+out to z = 1.4. However, it is still unclear when radio-
+mode feedback was established in galaxy clusters and
+how the fraction of clusters with well-regulated AGN
+feedback has evolved out to high redshifts. The next
+generation X-ray observatories will target this question
+by probing the ICM in the most distant clusters, with
+the ability to detect cluster emission out to z ∼ 2 − 3
+(Barret et al. 2020). With many more systems, we will
+be able to get a better handle on the evolution of radio-
+mode feedback and the AGN duty cycle in high redshift
+clusters. For now, at z = 1.401, SPT0607 provides the
+furthest constraint on the onset of radio-mode feedback
+in cool core clusters.
+4.2. Star Formation in BCGs at High Redshift
+Star formation in the BCGs in cool core clusters is a
+critical piece of the AGN feedback process as it acts as
+a probe of the balance between heating by AGN feed-
+back and cooling in the ICM. Various works have found
+that both the star formation rate and specific star for-
+mation rate of BCGs increase as a function of increasing
+redshift (e.g. Webb et al. 2015; McDonald et al. 2016b;
+Bonaventura et al. 2017). However, the nature of star
+forming BCGs seems to have changed with redshift. In
+particular, McDonald et al. (2016b) found that there
+was a transition in the fuel supply of the BCG, namely
+that high-redshift clusters out to z ∼ 1.2 with highly
+star forming BCGs were almost always disturbed clus-
+ters. This suggests that gas-rich mergers are responsible
+for runaway cooling and star formation in high-redshift
+systems, rather than cooling flows from a lack of heat-
+ing from AGN feedback, as was recently observed in the
+z ∼ 1.7 system SpARCS1049 (Hlavacek-Larrondo et al.
+2020).
+However, at low redshifts, star forming BCGs
+are predominantly found in relaxed systems, indicating
+that star formation in BCGs at low redshifts is com-
+monly driven by cooling of the ICM and regulated by
+AGN feedback. With multi-wavelength observations of
+SPT0607, we have found that this high-redshift, relaxed
+cluster hosts a BCG with very little star formation. The
+BCG also shows no noticeable morphological features in
+the 3-band HST images that suggest any recent merg-
+ers of interactions. These findings thus agree with the
+idea of a transitioning fuel supply for BCG star forma-
+tion at high redshift, where the majority of the fuel for
+star formation in high-redshift systems comes from gas
+rich mergers as clusters are assembling. SPT0607 sup-
+ports this picture out to z ∼ 1.4 and suggests that the
+early onset of AGN feedback provides sufficient heating
+to offset direct cooling from the ICM into stars at high
+redshift.
+5. SUMMARY
+We have presented a multi-wavelength analysis of one
+of the most distant SPT-selected clusters, SPT0607 at
+a redshift of z = 1.401. Through analysis of Chandra
+X-ray data, we found that SPT0607 has a strong cool
+core, as evidenced by both an increase in central gas
+density and a low entropy core as measured from the
+X-ray peak. These results follow from our conservative
+assumption of an isothermal temperature profile; in re-
+ality, we expect the central temperature of SPT0607 to
+drop in the center, which gives an even lower entropy
+core when assumed.
+As shown in Figure 1, the core of SPT0607 is co-
+incident with the BCG, which harbors a radio jet de-
+tected with ATCA at 2.1 GHz. Despite having a dense
+and cool core, we measure a star formation rate in the
+BCG of SPT0607 of SFR[O ii] = 1.7+1.0
+−0.6 M⊙ yr−1 using
+measurements of the [O ii] emission line from optical
+spectroscopy with the LDSS-3C instrument on the 6.5m
+Magellan Clay telescope.
+This star formation rate is
+roughly 1% of the expected mass cooling rate of the
+ICM of
+˙Mcool = 100+90
+−60 M⊙ yr−1 from our X-ray mea-
+surements. Similarly, we measure a cavity power from
+the radio jet of Pcav = 3.2+2.1
+−1.3 × 1044 erg s−1, which
+is consistent with the X-ray cooling luminosity.
+This
+indicates that the BCG in SPT0607 is providing radio-
+mode feedback to offset the cooling from the ICM. This
+phenomenon is commonplace at low redshift, but as one
+of the most distant clusters known to date, the regula-
+tion of cooling and AGN feedback in SPT0607 gives the
+strongest constraints on the onset of radio-mode AGN
+feedback in galaxy clusters to date.
+The South Pole Telescope program is supported by
+the National Science Foundation (NSF) through grants
+PLR-1248097 and OPP-1852617.
+Partial support is
+also provided by the NSF Physics Frontier Center grant
+
+A Multi-Wavelength View of SPT-CL J0607-4448
+11
+PHY-1125897 to the Kavli Institute of Cosmological
+Physics at the University of Chicago, the Kavli Foun-
+dation, and the Gordon and Betty Moore Foundation
+through grant GBMF#947 to the University of Chicago.
+Argonne National Laboratory’s work was supported by
+the U.S. Department of Energy, Office of Science, Of-
+fice of High Energy Physics, under contract DE-AC02-
+06CH11357. The Melbourne group acknowledges sup-
+port from the Australian Research Council’s Discovery
+Projects scheme (DP200101068).
+All of the HST data used in this paper can be found
+in MAST: 10.17909/e40m-z102.
+Facilities: CXO, HST, Magellan, ATCA, NSF/US
+Department of Energy 10m South Pole Telescope (SPT-
+SZ)
+Software:
+CIAO (Fruscione et al. 2006), XSPEC
+(Arnaud 1996), CASA (McMullin et al. 2007), STAR-
+BURST99 (Leitherer et al. 1999), Astropy (Astropy Col-
+laboration et al. 2013, 2018), Matplotlib (Hunter 2007),
+NumPy (van der Walt et al. 2011)
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+Particle identification with the Belle II calorimeter
+using machine learning
+Abtin Narimani Charan
+Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany
+E-mail: [email protected]
+Abstract.
+I present an application of a convolutional neural network (CNN) to separate muons
+and pions in the Belle II electromagnetic calorimeter (ECL). The ECL is designed to measure
+the energy deposited by charged and neutral particles. It also provides important contributions
+to the particle identification (PID) system. Identification of low-momenta muons and pions in
+the ECL is crucial if they do not reach the outer muon detector. Track-seeded cluster energy
+images provide the maximal possible information.
+The shape of the energy depositions for
+muons and pions in the crystals around an extrapolated track at the entering point of the ECL
+is used together with crystal positions in θ − φ plane and transverse momentum of the track to
+train a CNN. The CNN exploits the difference between the dispersed energy depositions from
+pion hadronic interactions and the more localized muon electromagnetic interactions. Using
+simulation, the performance of the CNN algorithm is compared with other PID methods at
+Belle II which are based on track-matched clustering information. The results show that the
+CNN PID method improves muon-pion separation in low momentum.
+1. Introduction
+The Belle II experiment at the SuperKEKB e+e− collider belongs to the next generation of B
+factories and is the upgraded version of the Belle experiment. Among the main goals of the
+Belle II experiment there are the search of New Physics and the high precision measurements of
+the Standard Model parameters in the flavour sector [1]. Belle II intends to collect data samples
+with an integrated luminosity of 50 ab−1 at predominantly the Υ(4S) resonance. SuperKEKB’s
+goal for instantaneous luminosity is 6 × 1035 cm−2 s−1, which is 40 times larger than that of
+KEKB. This luminosity can be achieved due to a significant decrease in the beam sizes by a
+factor of 20 at the interaction point based on the nano-beam collision scheme and by doubling the
+beam currents in both rings [2, 3]. With the above-mentioned improvements, Belle II encounters
+higher beam background level. The physics reach of Belle II can be enhanced if a better muon-
+pion separation can be achieved.
+A better muon-pion separation for low-momenta tracks is
+advantageous for B physics with leptonic τ decays, since this can reduce the number of pions
+misidentified as muons at low momentum. It is crucial to rely on the information in the ECL
+because low-momentum muons (p ≲ 0.6 GeV/c) cannot reach the dedicated, outermost detector
+K0
+L-and-muon detector (KLM).
+2. The ECL
+The ECL consists of 8736 thallium-doped caesium iodide CsI(Tl) crystals projected to the
+vicinity of the interaction point. It is immersed in a 1.5 T magnetic field and composed of
+arXiv:2301.11654v1  [hep-ex]  27 Jan 2023
+
+a barrel and two endcaps with polar angle coverage of 12.4◦ – 155.1◦. The dimension of each
+crystal is ∼ 5×5 cm2 with a length of 30 cm corresponding to 16.2 X0 (radiation length). The
+barrel has 6624 crystals positioned in 46 rings of crystals distributed in the θ-plane and each
+ring consists of 144 crystals in the φ-plane [1, 3]. In this study, tracks are extrapolated into the
+ECL. Then, at the entry point of the track into the ECL, a window of 7×7 crystals with the
+measured deposited energy therein is selected. Since the number of crystals in the barrel in each
+θ-plane is equal to 144, the 7×7 crystal images are symmetrical. Typical patterns of energy
+depositions for muons and pions are shown in figure 1. Muons and pions both deposit energy
+by ionization in the matter. Additionally, pions can undergo hadronic interactions. Therefore,
+crystal images of muons are generally more localized than pions.
+Figure 1. Patterns of energy depositions for muons (blue) and pions (red) inside the ECL
+barrel.
+3. Charged particle identification at Belle II
+There are two charged PID methods available in the Belle II analysis software framework [4]. The
+first method is standard PID which is based on a combination of measurements from different
+sub-detectors. For each charged particle hypothesis (i) in each PID system, a likelihood LPID
+i
+is determined. It is a function of the probability density function parameters for a given set
+of observables. These likelihoods can be used to construct a combined likelihood ratio for a
+particular sub-detector. A binary likelihood ratio for the ECL system between muon and pion
+is defined as RECL
+µ/π =
+LECL
+µ
+LECL
+µ
++LECL
+π
+which is used as benchmark for comparison in the study, and
+is indicated as default. The standard PID in the ECL (LECL) defines a univariate likelihood as
+a function of E/p i.e., the ratio of reconstructed cluster energy over the momentum. The E/p
+distribution is not very powerful for muon-pion separation specifically for low momentum tracks
+(figure 2). The second method is based on boosted decision trees (BDTs) [5]. It uses the shower-
+shape information in the ECL combined with likelihood information from other sub-detectors
+to train BDTs.
+4. Convolutional neural network (CNN)
+A convolutional neural network (CNN) is a type of deep neural network which is used to recognize
+visual patterns from pixel images [6]. In this study, two CNNs are trained with 7×7 pixel images
+of muons and pions together with crystal positions and transverse momentum of the track in the
+laboratory frame (each pixel is a ∼ 5×5 cm2 crystal). Separate CNNs are trained for positive and
+negative charged tracks. This is due to the geometry of the ECL i.e. the direction of the crystals.
+Approximately 1 million single muon and pion candidates are generated with a flat distribution
+of transverse momentum between 0.2 and 1.0 GeV/c. Each track is first reconstructed in the
+inner tracking detectors, and then extrapolated into the ECL with Geant4 [7].
+
+0.10
+GeV
+0.05 g
+Energ
+10.00
+0.10
+[GeV]
+0.05 g
+Ener
+0.00Figure 2. The distribution of the ratio of cluster energy over momentum (E/p) for simulated
+single track candidates of muons (blue) and pions (red) inside the ECL barrel for low (left) and
+high (right) momentum range.
+4.1. Inputs, pre-processing, and training
+There are two types of inputs for the CNN. One is the energy depositions in the 7×7 pixel images
+and the other is a set of inputs which are fed after the convolutional layers that includes pT , θID,
+and φID of the extrapolated tracks. The θID and φID represent integer numbers corresponding to
+the location of the crystal in the ECL. The energies in the pixel images are given as they are, i.e.,
+without any scaling applied since they already have small values. However, there are very large
+values in a few pixel images of pions which have energies more than 1 GeV. This is due to pion
+inelastic interaction with nuclei producing protons. These large values are replaced with 1 GeV.
+Since there are very few of these pixels, this adjustment is negligible since it affects the standard
+deviation and mean of the energy depositions by 0.09 % and 0.005 %, respectively. A threshold
+value of 1 MeV on the energy depositions in the pixels is applied, so that pixels below this
+threshold are assigned zero energy. The pT is already in the range of 0.2 – 1.0 GeV/c, therefore
+no scaling is applied. The θID and φID are used as categorical variables which are implemented
+as an embedding in the network. An embedding is a distributed representation for categorical
+variables where each category is mapped to a distinct vector that a neural network can learn
+during training. The number of simulated events is identical for signal and background. The
+equal size of the sample is important to avoid bias against a particular type of particle or charge.
+The CNN includes two parts. In the first part, the 7×7 pixel images of muons and pions are
+used as inputs for a convolutional layer. During convolution, a window of 3×3, called kernel,
+goes through each image. A padding and stride of (1, 1) is used for the images. The padding
+adds one pixel with value zero on the edge of the image which is beneficial to capture more
+information on the edges. The stride refers to the amount of kernel movement over the image.
+The feature map is set to 64. The second part of the training involves a feed-forward neural
+network (FNN). In this part, the results from the convolutional layer must be flattened and
+added on top of pT , θID, and φID. The number of neurons in the first and second layer of FNN
+are 3295 and 128, respectively. The dropout layer with a value of 0.1 is used between the first
+and second layer of the FNN. An Adam optimizer is used during the training with a learning rate
+of 0.001. The loss used in this study is CrossEntropy which is suitable for binary classification
+problems. The number of epochs is set to 100. The model is saved based on the lowest validation
+loss value in the corresponding epoch. A schematic of the network is shown in figure 3.
+4.2. Performance
+The performance of the CNN is evaluated on a test dataset with an equal number of muons
+and pions. The test dataset is generated and reconstructed with the exact same conditions as
+the training dataset. The CNN is compared with two other methods (default and BDT). The
+performance of all methods for both charged tracks in a range of transverse momentum (0.52 ≤
+
+1.2 
+×104
+Belle ll Simulation
+Belle ll Simulation
+从t
+3.5
+ut
+1.0
+t
+3.0
+0.8
+2.5
+nts
+0.2 ≤ p≤ 0.6 GeV/c
+nts
+p > 0.9 GeV/c
+ECL barrel
+ECL barrel
+00.6
+02.0
+Eve
+0.4
+1.0
+0.2
+0.5
+0.0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+:8.0
+0.2
+0.4
+0.6
+0.8
+1.0
+1.2
+E/p (c)
+E/p (c)Figure 3. Neural network architecture. See text for details.
+pT < 0.76 GeV/c) are shown in figure 4. The µ efficiency is defined as the ratio of the number
+of correctly identified muons over the total number of muons. The π fake rate is defined as
+the ratio of the number of pions identi���ed as muons over the total number of pions. The CNN
+method outperforms the other methods with AUC (Area Under the Curve) scores of 0.836 and
+0.841 for positive and negative charged tracks, respectively.
+Figure 4. Comparison of CNN, BDT, and default PID. The left and right plots show positive
+and negative charged tracks, respectively. The value in front of each method shows the area
+under the curve (AUC).
+There are cases (mostly pions) that the software framework does not match a track with
+a cluster in the ECL. Since the CNN method does not rely on clustering, a comparison is
+made among all tracks and tracks with and without matched cluster in the inference phase. The
+comparison is shown in figure 5 in which the blue ROC curves, representing the CNN (all tracks),
+can be considered as an average between the red and green curves, which represent tracks with
+and without matched clusters, respectively. The large difference between red and green ones is
+due to statistics. The green ROC curve is roughly 2.8% and 3.5% of the test dataset in the pT
+range of 0.52 – 0.76 GeV/c for positive and negative charged tracks, respectively.
+Due to the higher beam background level at Belle II, a higher minimal energy threshold for
+crystals may reduce the pile-up from beam background. In order to check the robustness of
+the CNN method against different levels of beam background, different energy thresholds for
+crystals of 0, 1, 2, 5, and 8 MeV are tested and the results are shown in figure 6 (zero means no
+threshold). Results show the CNN is robust against the choice of different energy thresholds.
+
+FC1
+Inputs
+Binary
+00
+FC2
+PT
+0000
+Classification
+μ
+元
+O1D
+Problem
+ΦID
+Dropout
+P(元)
+0.1
+O
+P(μ)
+..000
+00
+Energy
+Convolutional
+layers
+deposition1.0
+1.0
+0.8
+0.8
+Belle ll Simulation
+Belle ll Simulation
+iency
+0.52 ≤pr< 0.76 GeV/c
+0.52 ≤pr< 0.76 GeV/c
+0.6
+ECL barrel
+0.6
+ECL barrel
+Efficie
+0.4
+CNN (0.836)
+CNN (0.841)
+0.2
+0.2
+BDT (0.815)
+BDT (0.805)
+Default (0.725)
+Default (0.702)
+0.0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
++ Fake rate
+π- Fake rateFigure 5. Comparison of CNN performance for all tracks, tracks with and without matched
+cluster. The left and right plots show positive and negative charged tracks, respectively.
+Figure 6. Comparison of CNN performance for different thresholds. The left and right plots
+show positive and negative charged tracks, respectively.
+5. Summary and outlook
+This study shows that by using patterns of energy depositions in the ECL, muon-pion separation
+can be improved for low-momenta tracks. The pion fake rate is 4.2% and 6.7% larger at a typical
+working point of 90% muon identification efficiency for positive and negative charged tracks,
+respectively. The CNN method is available in the Belle II analysis software framework and will
+be integrated as part of the standard Belle II reconstruction software. This study can be extended
+to include additional low-level ECL crystal information, e.g., pulse-shape discrimination [8, 9]
+which is useful to separate hadronic and electromagnetic interactions. In order to validate the
+CNN method on data, clean samples of muons and pions are selected using e+e− → µ+µ−γ and
+D∗ + → D0 [→ K+ π−] π+, respectively. These results are underway.
+References
+[1] Kou E et al. 2019 The Belle II Physics Book Prog. Theor. Exp. Phys. 2019 123C01
+[2] Akai K, Furukawa K and Koiso H (SuperKEKB) 2018 SuperKEKB Collider Nucl. Instrum. Meth. A 907,
+188–99
+[3] Abe T et al. 2010 Belle II Technical Design Report 1011.0352
+[4] Kuhr T, Pulvermacher C, Ritter M, Hauth T and Braun N 2019 The Belle II Core Software Comput. Softw.
+Big Sci. 3 1
+[5] Milesi M, Tan J and Urquijo P 2020 Lepton identification in Belle II using observables from the electromagnetic
+calorimeter and precision trackers EPJ Web Conf. 245 06023
+[6] Bishop C M 2006 Pattern Recognition and Machine Learning (Singapore: Springer Science+Business Media,
+LLC)
+[7] Agostinelli S et al. (GEANT4) 2003 GEANT4–a simulation toolkit Nucl. Instrum. Meth. A 506 250–303
+[8] Longo S and Roney J M 2018 Hadronic vs. electromagnetic pulse shape discrimination in CsI(Tl) for high
+energy physics experiments JINST 13 P03018
+[9] Longo S et al. 2020 CsI(Tl) pulse shape discrimination with the Belle II electromagnetic calorimeter as a novel
+method to improve particle identification at electron-positron colliders Nucl. Instrum. Meth. A 982 164562
+
+1.0
+1.0
+0.8
+0.8
+Belle ll Simulation
+Belle ll Simulation
+iency
+iency
+0.52 ≤pr< 0.76 GeV/c
+0.52 ≤pr< 0.76 GeV/c
+0.6
+ECL barrel
+0.6
+ECL barrel
+0.4
+CNN (all tracks)
+CNN (all tracks)
+0.2
+0.2
+CNN (tracks with matched clusters)
+CNN (tracks with matched clusters)
+CNN (tracks without matched clusters)
+CNN (tracks without matched clusters)
+0.0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
++ Fake rate
+π- Fake rate1.0
+1.0
+0.8
+0.8
+Belle ll Simulation
+Belle ll Simulation
+iency
+0.28 ≤pr < 1.00 GeV/c
+0.28 ≤pt< 1.00 GeV/c
+0.6
+ECL barrel
+0.6
+ECL barrel
+CNN (thr=0 MeV)
+CNN (thr=O MeV)
+CNN (thr=1 MeV)
+CNN (thr=1 MeV)
+CNN (thr=2 MeV)
+CNN (thr=2 MeV)
+0.2
+0.2
+CNN (thr=5 MeV)
+CNN (thr=5 MeV)
+CNN (thr=8 MeV)
+CNN (thr=8 MeV)
+0.0
+0.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+π+ Fake rate
+π- Fake rate

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+page_content='Particle identification with the Belle II calorimeter  using machine learning  Abtin Narimani Charan  Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany  E-mail: abtin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='narimani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='charan@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='de  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  I present an application of a convolutional neural network (CNN) to separate muons  and pions in the Belle II electromagnetic calorimeter (ECL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The ECL is designed to measure  the energy deposited by charged and neutral particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' It also provides important contributions  to the particle identification (PID) system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Identification of low-momenta muons and pions in  the ECL is crucial if they do not reach the outer muon detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Track-seeded cluster energy  images provide the maximal possible information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  The shape of the energy depositions for  muons and pions in the crystals around an extrapolated track at the entering point of the ECL  is used together with crystal positions in θ − φ plane and transverse momentum of the track to  train a CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The CNN exploits the difference between the dispersed energy depositions from  pion hadronic interactions and the more localized muon electromagnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Using  simulation, the performance of the CNN algorithm is compared with other PID methods at  Belle II which are based on track-matched clustering information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The results show that the  CNN PID method improves muon-pion separation in low momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Introduction  The Belle II experiment at the SuperKEKB e+e− collider belongs to the next generation of B  factories and is the upgraded version of the Belle experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Among the main goals of the  Belle II experiment there are the search of New Physics and the high precision measurements of  the Standard Model parameters in the flavour sector [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Belle II intends to collect data samples  with an integrated luminosity of 50 ab−1 at predominantly the Υ(4S) resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' SuperKEKB’s  goal for instantaneous luminosity is 6 × 1035 cm−2 s−1, which is 40 times larger than that of  KEKB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' This luminosity can be achieved due to a significant decrease in the beam sizes by a  factor of 20 at the interaction point based on the nano-beam collision scheme and by doubling the  beam currents in both rings [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' With the above-mentioned improvements, Belle II encounters  higher beam background level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The physics reach of Belle II can be enhanced if a better muon-  pion separation can be achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  A better muon-pion separation for low-momenta tracks is  advantageous for B physics with leptonic τ decays, since this can reduce the number of pions  misidentified as muons at low momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' It is crucial to rely on the information in the ECL  because low-momentum muons (p ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6 GeV/c) cannot reach the dedicated, outermost detector  K0  L-and-muon detector (KLM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The ECL  The ECL consists of 8736 thallium-doped caesium iodide CsI(Tl) crystals projected to the  vicinity of the interaction point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' It is immersed in a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='5 T magnetic field and composed of  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='11654v1  [hep-ex]  27 Jan 2023  a barrel and two endcaps with polar angle coverage of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='4◦ – 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='1◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The dimension of each  crystal is ∼ 5×5 cm2 with a length of 30 cm corresponding to 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2 X0 (radiation length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The  barrel has 6624 crystals positioned in 46 rings of crystals distributed in the θ-plane and each  ring consists of 144 crystals in the φ-plane [1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' In this study, tracks are extrapolated into the  ECL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Then, at the entry point of the track into the ECL, a window of 7×7 crystals with the  measured deposited energy therein is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Since the number of crystals in the barrel in each  θ-plane is equal to 144, the 7×7 crystal images are symmetrical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Typical patterns of energy  depositions for muons and pions are shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Muons and pions both deposit energy  by ionization in the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Additionally, pions can undergo hadronic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Therefore,  crystal images of muons are generally more localized than pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Patterns of energy depositions for muons (blue) and pions (red) inside the ECL  barrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Charged particle identification at Belle II  There are two charged PID methods available in the Belle II analysis software framework [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The  first method is standard PID which is based on a combination of measurements from different  sub-detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' For each charged particle hypothesis (i) in each PID system, a likelihood LPID  i  is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' It is a function of the probability density function parameters for a given set  of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' These likelihoods can be used to construct a combined likelihood ratio for a  particular sub-detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' A binary likelihood ratio for the ECL system between muon and pion  is defined as RECL  µ/π =  LECL  µ  LECL  µ  +LECL  π  which is used as benchmark for comparison in the study, and  is indicated as default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The standard PID in the ECL (LECL) defines a univariate likelihood as  a function of E/p i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=', the ratio of reconstructed cluster energy over the momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The E/p  distribution is not very powerful for muon-pion separation specifically for low momentum tracks  (figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The second method is based on boosted decision trees (BDTs) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' It uses the shower-  shape information in the ECL combined with likelihood information from other sub-detectors  to train BDTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Convolutional neural network (CNN)  A convolutional neural network (CNN) is a type of deep neural network which is used to recognize  visual patterns from pixel images [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' In this study, two CNNs are trained with 7×7 pixel images  of muons and pions together with crystal positions and transverse momentum of the track in the  laboratory frame (each pixel is a ∼ 5×5 cm2 crystal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Separate CNNs are trained for positive and  negative charged tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' This is due to the geometry of the ECL i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' the direction of the crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  Approximately 1 million single muon and pion candidates are generated with a flat distribution  of transverse momentum between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Each track is first reconstructed in the  inner tracking detectors, and then extrapolated into the ECL with Geant4 [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='10  GeV  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='05 g  Energ  10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='00  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='10  [GeV]  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='05 g  Ener  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='00Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The distribution of the ratio of cluster energy over momentum (E/p) for simulated  single track candidates of muons (blue) and pions (red) inside the ECL barrel for low (left) and  high (right) momentum range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Inputs, pre-processing, and training  There are two types of inputs for the CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' One is the energy depositions in the 7×7 pixel images  and the other is a set of inputs which are fed after the convolutional layers that includes pT , θID,  and φID of the extrapolated tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The θID and φID represent integer numbers corresponding to  the location of the crystal in the ECL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The energies in the pixel images are given as they are, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=',  without any scaling applied since they already have small values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' However, there are very large  values in a few pixel images of pions which have energies more than 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' This is due to pion  inelastic interaction with nuclei producing protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' These large values are replaced with 1 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  Since there are very few of these pixels, this adjustment is negligible since it affects the standard  deviation and mean of the energy depositions by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='09 % and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='005 %, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' A threshold  value of 1 MeV on the energy depositions in the pixels is applied, so that pixels below this  threshold are assigned zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The pT is already in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2 – 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0 GeV/c, therefore  no scaling is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The θID and φID are used as categorical variables which are implemented  as an embedding in the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' An embedding is a distributed representation for categorical  variables where each category is mapped to a distinct vector that a neural network can learn  during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The number of simulated events is identical for signal and background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The  equal size of the sample is important to avoid bias against a particular type of particle or charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  The CNN includes two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' In the first part, the 7×7 pixel images of muons and pions are  used as inputs for a convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' During convolution, a window of 3×3, called kernel,  goes through each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' A padding and stride of (1, 1) is used for the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The padding  adds one pixel with value zero on the edge of the image which is beneficial to capture more  information on the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The stride refers to the amount of kernel movement over the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  The feature map is set to 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The second part of the training involves a feed-forward neural  network (FNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' In this part, the results from the convolutional layer must be flattened and  added on top of pT , θID, and φID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The number of neurons in the first and second layer of FNN  are 3295 and 128, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The dropout layer with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='1 is used between the first  and second layer of the FNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' An Adam optimizer is used during the training with a learning rate  of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The loss used in this study is CrossEntropy which is suitable for binary classification  problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The number of epochs is set to 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The model is saved based on the lowest validation  loss value in the corresponding epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' A schematic of the network is shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Performance  The performance of the CNN is evaluated on a test dataset with an equal number of muons  and pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The test dataset is generated and reconstructed with the exact same conditions as  the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The CNN is compared with two other methods (default and BDT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The  performance of all methods for both charged tracks in a range of transverse momentum (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='52 ≤  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2  ×104  Belle ll Simulation  Belle ll Simulation  从t  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='5  ut  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  t  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='8  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='5  nts  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2 ≤ p≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6 GeV/c  nts  p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='9 GeV/c  ECL barrel  ECL barrel  00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6  02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  Eve  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='4  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='5  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2  :8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='4  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2  E/p (c)  E/p (c)Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  pT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='76 GeV/c) are shown in figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The µ efficiency is defined as the ratio of the number  of correctly identified muons over the total number of muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The π fake rate is defined as  the ratio of the number of pions identified as muons over the total number of pions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The CNN  method outperforms the other methods with AUC (Area Under the Curve) scores of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='836 and  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='841 for positive and negative charged tracks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Comparison of CNN, BDT, and default PID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The left and right plots show positive  and negative charged tracks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The value in front of each method shows the area  under the curve (AUC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  There are cases (mostly pions) that the software framework does not match a track with  a cluster in the ECL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Since the CNN method does not rely on clustering, a comparison is  made among all tracks and tracks with and without matched cluster in the inference phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The  comparison is shown in figure 5 in which the blue ROC curves, representing the CNN (all tracks),  can be considered as an average between the red and green curves, which represent tracks with  and without matched clusters, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The large difference between red and green ones is  due to statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The green ROC curve is roughly 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='8% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='5% of the test dataset in the pT  range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='52 – 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='76 GeV/c for positive and negative charged tracks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  Due to the higher beam background level at Belle II, a higher minimal energy threshold for  crystals may reduce the pile-up from beam background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' In order to check the robustness of  the CNN method against different levels of beam background, different energy thresholds for  crystals of 0, 1, 2, 5, and 8 MeV are tested and the results are shown in figure 6 (zero means no  threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Results show the CNN is robust against the choice of different energy thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  FC1  Inputs  Binary  00  FC2  PT  0000  Classification  μ  元  O1D  Problem  ΦID  Dropout  P(元)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='1  O  P(μ)  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='.000  00  Energy  Convolutional  layers  deposition1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='8  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='8  Belle ll Simulation  Belle ll Simulation  iency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='52 ≤pr< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='76 GeV/c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='52 ≤pr< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='76 GeV/c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6  ECL barrel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6  ECL barrel  Efficie  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='4  CNN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='836)  CNN (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='841)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2  BDT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='815)  BDT (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='805)  Default (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='725)  Default (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='702)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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+page_content='6  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='8  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0  + Fake rate  π- Fake rateFigure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Comparison of CNN performance for all tracks, tracks with and without matched  cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The left and right plots show positive and negative charged tracks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Comparison of CNN performance for different thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The left and right plots  show positive and negative charged tracks, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Summary and outlook  This study shows that by using patterns of energy depositions in the ECL, muon-pion separation  can be improved for low-momenta tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The pion fake rate is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='2% and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='7% larger at a typical  working point of 90% muon identification efficiency for positive and negative charged tracks,  respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' The CNN method is available in the Belle II analysis software framework and will  be integrated as part of the standard Belle II reconstruction software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' This study can be extended  to include additional low-level ECL crystal information, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=', pulse-shape discrimination [8, 9]  which is useful to separate hadronic and electromagnetic interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' In order to validate the  CNN method on data, clean samples of muons and pions are selected using e+e− → µ+µ−γ and  D∗ + → D0 [→ K+ π−] π+, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' These results are underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  References  [1] Kou E et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' 2019 The Belle II Physics Book Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' 2019 123C01  [2] Akai K, Furukawa K and Koiso H (SuperKEKB) 2018 SuperKEKB Collider Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' A 907,  188–99  [3] Abe T et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' 2010 Belle II Technical Design Report 1011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='0352  [4] Kuhr T, Pulvermacher C, Ritter M, Hauth T and Braun N 2019 The Belle II Core Software Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Softw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='  Big Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' 3 1  [5] Milesi M, Tan J and Urquijo P 2020 Lepton identification in Belle II using observables from the electromagnetic  calorimeter and precision trackers EPJ Web Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' 245 06023  [6] Bishop C M 2006 Pattern Recognition and Machine Learning (Singapore: Springer Science+Business Media,  LLC)  [7] Agostinelli S et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' (GEANT4) 2003 GEANT4–a simulation toolkit Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' A 506 250–303  [8] Longo S and Roney J M 2018 Hadronic vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' electromagnetic pulse shape discrimination in CsI(Tl) for high  energy physics experiments JINST 13 P03018  [9] Longo S et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' 2020 CsI(Tl) pulse shape discrimination with the Belle II electromagnetic calorimeter as a novel  method to improve particle identification at electron-positron colliders Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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+page_content=' A 982 164562  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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+page_content='8  Belle ll Simulation  Belle ll Simulation  iency  iency  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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+page_content='76 GeV/c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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+page_content='76 GeV/c  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
+page_content='6  ECL barrel  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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+page_content='0  + Fake rate  π- Fake rate1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PtFJT4oBgHgl3EQf2C2D/content/2301.11654v1.pdf'}
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Q9FPT4oBgHgl3EQfpTVX/content/tmp_files/2301.13137v1.pdf.txt

@@ -0,0 +1,448 @@
+Multiple pion pair production in a Regge-based model∗
+Rainer Schicker
+Physikalisches Institut, University Heidelberg, Heidelberg
+in coll. with Laszlo Jenkovszky
+Bogolyubov ITP, National Academy of Sciences of Ukraine, Kiev
+Received January 31, 2023
+Central diffractive event topologies at the LHC energies can be identi-
+fied by two different approaches. First, the forward scattered protons can
+be measured in Roman pots. Second, a veto on hadronic activity away
+from midrapidity can be imposed to define a double-gap topology. Such
+a double-gap topology trigger has been implemented by the ALICE col-
+laboration in Run 1 and Run 2 of the LHC. The analysis of these events
+allows to determine the charged-particle multiplicity within the acceptance.
+The excellent particle identification capabilities of ALICE allows to study
+two-track events both in the pion and kaon sector. Events with measured
+charged particle multiplicity larger than two can arise from multiple pair
+production. A Regge-based approach for modeling such multiple pair pro-
+duction is presented.
+1. Introduction
+Double-Pomeron fusion at hadron colliders results in a double-gap event
+topology. Such a topology is defined by hadronic activity at or close to
+midrapidity, and the absence thereof away from midrapidity. The multi-
+plicity distribution of such double-gap events has been measured in the
+ALICE central barrel. To better understand such multiplicity distributions
+we present here a Regge-based approach for multiple pion pair production
+in double-Pomeron events. This model is based on a Dual Amplitude with
+Mandelstam Analyticity (DAMA) [1]. In this approach, the production of
+multiple pairs can be modeled by including a Pomeron-Pomeron-Reggeon
+and a triple-Pomeron coupling. The amplitude at Pomeron level within his
+DAMA formulation is given, and the resulting mass distributions for double
+pion and double b-resonance production are shown.
+∗ Presented at ”Diffraction and Low-x 2022”, Corigliano Calabro (Italy), September
+24-30, 2022.
+(1)
+arXiv:2301.13137v1  [hep-ph]  30 Jan 2023
+
+2
+schicker˙diff2022
+printed on January 31, 2023
+2. Multiplicity distribution of double-gap events
+The charged-particle multiplicity in the ALICE central barrel has been
+analyzed in LHC Run 1 for both minimum bias and double-gap events [2].
+0
+2
+4
+6
+8
+10
+)
+ch
+(n
+MinBias
+N
+)
+ch
+(n
+DG
+N
+) =
+ch
+(n
+DG
+R
+-6
+10
+-5
+10
+-4
+10
+-3
+10
+-2
+10
+Data
+Tuned Pythia6
+Tuned Phojet
+Pythia6
+Phojet
+Phojet w/o CD
+Phojet CD only
+ch
+N
+0
+2
+4
+6
+8
+10
+Ratio MC to Data
+-1
+10
+1
+10
+Fig. 1:
+Double-gap probability in ALICE central barrel as function of
+charged-particle multiplicity (Figure taken from Ref. [2]).
+In Fig. 1, the probability of being a double-gap event is shown as func-
+tion of the charged-particle multiplicity Nch in the ALICE central barrel.
+The ALICE data are shown in black circles, whereas the results from Monte
+Carlo generators are shown in different colors. These probabilities clearly
+show a maximum at Nch=1 and Nch=2, demonstrating that double-Pomeron
+events are dominated by very low multiplicities as compared to minimum
+bias events. As indicated in this figure, none of the tested generators shows
+reasonable agreement with the data. This discrepancy between the ALICE
+measured double-gap events and the prediction of the tested generators
+motivates the development of a model which can be used to analyze unlike-
+sign two-track events resulting from single resonance decays, as well as the
+higher-multiplicity events stemming from the decays of multiple resonances.
+3. A Regge model for double-Pomeron events
+The model for Pomeron-Pomeron-induced events presented in the fol-
+lowing is based on the DAMA approach. Pomeron-induced single-resonance
+production has been presented in our previous studies [3,4]. Here, we extend
+this DAMA approach to the production of multiple resonances.
+
+schicker˙diff2022
+printed on January 31, 2023
+3
+1
+-
+amplitude
+subdiagram
+Fig. 2: Amplitude at hadron level (left), and Pomeron subdiagram (right).
+In Fig. 2, the amplitude for Pomeron-induced single-resonance produc-
+tion at hadron level is shown on the left.
+The subdiagram on the right
+represents the amplitude for Pomeron-Pomeron → resonance. The cross
+section at hadron level is derived by convoluting the subdiagram cross sec-
+tion with the Pomeron flux of the proton F P
+prot(t, ξ) defined by
+F P
+prot(t, ξ) = 9β2
+0
+4π2 [F1(t)]2ξ1−2α(t),
+(1)
+with F1(t) the elastic form factor, and α(t) the Pomeron trajectory [3].
+In the DAMA approach, multiple-resonance production can be modeled
+by introducing a Pomeron-Pomeron-Reggeon (PPR) coupling with subse-
+quent splitting of the intermediate Reggeon into the two final-state Reggeons.
+Alternatively, the same final state can be formed by a triple-Pomeron (PPP)
+coupling with the intermediate Pomeron decaying into the two Reggeons.
+1
+αP
+t1
+αP
+t2
+g1
+g2
+-
+˜s
+αR(˜s)
+?
+˜t
++
+˜S1(M2
+1 )
+˜S2(M2
+2 )
+(˜s=t1+t2)
+g3
+g4
+αP(˜s)
+˜S1(M2
+1 )
+˜S2(M2
+2 )
+Fig. 3: Subdiagram for PPR amplitude (left), and PPP amplitude (right).
+The DAMA amplitude for the subdiagram shown in Fig. 3 is given by
+
+00000000000000004
+schicker˙diff2022
+printed on January 31, 2023
+APP→ ˜S1 ˜S2(˜s, ˜t, M2
+1 , M2
+2 )) =
+1
+�
+M2
+1 M2
+2
+�
+PPR,PPP
+�
+n
+gigjebα(˜t)
+n − α(˜s) ,
+(2)
+with the first summation over the two amplitudes of Fig. 3 defined by the
+PPR coupling with gi,gj=g1,g2, and the PPP coupling with gi,gj=g3,g4. The
+index n sums over the spins of the resonances of the intermediate trajectory
+which connects the vertices i, j. From this amplitude, the cross section at
+Pomeron level is derived by the optical theorem
+σt(˜s, M2
+1 , M2
+2 ) = ℑm A(˜s, ˜t=0, M2
+1 , M2
+2 ),
+(3)
+with the imaginary part of A(˜s, ˜t, M2
+1 , M2
+2 ) defined by αR(˜s) and αP(˜s) for
+the PPR and the PPP diagrams of Fig. 3, respectively.
+4. Reggeizing q¯q states in the light quark sector
+The final-state mesons derive from the decay of the meson resonances
+lying on the two Regge trajectories ˜S1 and ˜S2 as illustrated in Fig. 3. In or-
+der to be able to include mesonic bound states of different radial and orbital
+excitations, a unified description of q¯q bound states in the different flavour
+sectors is needed. Such a unified description of q¯q bound states including a
+confinement potential, a spin-orbit, a hyperfine and an annihilation inter-
+action is presented in Ref. [5]. The solutions for these q¯q bound states are
+given in spectroscopic notation n 2S+1LJ.
+spectr. notation n 2S+1LJ:
+- n radial quantum number
+- S spin
+- L orbital ang. momentum
+- J total ang. momentum
+n2S+1LJ
+mass
+PDG
+mass
+width
+Ref. [5]
+(PDG)
+(PDG)
+11S0
+150
+π
+140
+0
+11P1
+1220
+b1
+1230
+142
+11D2
+1680
+π2
+1672
+258
+11F3
+2030
+——
+——
+——
+11G4
+2330
+——
+——
+——
+Table 1: Masses and widths in MeV.
+In Table 1, masses are presented for the isovector channel in the light
+quark sector for the radial ground state for S,P,D,F and G-wave, and are
+compared to the values given by the Particle Data Group [6]. The S- and
+D-wave bound states calculated in Ref. [5] are identified with the π and the
+π2 states of mass 140 and 1672 MeV, respectively. The P-wave solution
+is associated to the known b1 state of mass 1230 MeV. No candidates for
+the predicted F- and G-wave bound states have so far been experimentally
+identified [6].
+
+schicker˙diff2022
+printed on January 31, 2023
+5
+5. Non-linear complex Regge trajectory
+The small but existing non-linear dependence of the spin of a resonance
+to its mass squared can be used to make a Regge trajectory α(M2) a complex
+entity with real and imaginary parts being related by a dispersion relation [7].
+Here, the real part is defined by the value of the spin, and the imaginary
+part is related to the decay width Γ by ℑmα(M2
+R) = Γ(MR)α
+′ MR, with α
+′
+denoting the derivative of the real part of the trajectory. In a simple model,
+the imaginary part is chosen as a sum of single threshold terms
+ℑm α(s)=
+�
+n
+cn(s−sn)1/2�s−sn
+s
+�|ℜe α(sn)|θ(s−sn).
+(4)
+In Eq. 4, the coefficients cn are fit parameters, and the parameters sn
+represent kinematical thresholds of decay channels.
+5.1. The (π, b)-trajectory
+A Regge trajectory, called the (π, b)-trajectory hereafter, is defined by
+the values of mass and width of the S, P and D-waves shown in Table 1.
+----------------
+----------------------
+----------------------------
+b
+b
+b
+π
+b1
+π2
+b3
+π4
+b5
+π
+b1
+π2
+s0
+s1
+Fig. 4: Real part (π, b)-trajectory on the left, width function Γ on the right.
+In Fig. 4 on the left, the three data points of the π, b1 and π2 states are
+shown by black points, and the non-linear fit by the blue line. On the right,
+the widths of the π, b1 and π2 states are shown by black points, and the fitted
+width function Γ by the blue line. The thresholds s0 and s1 used in the fit of
+Eq. 4 are shown in red. The thresholds s0=0.176 GeV2 and s1=1.27 GeV2
+are defined by the decays π2 →3π and b1 →K ¯Kπ, respectively. This fit of
+the (π, b)-trajectory predicts a b3 state with mass of 2090 MeV and width
+of 321 MeV, a π4 state with mass of 2437 MeV and width of 352 MeV, and
+a b5 state with mass of 2738 MeV and width of 371 MeV.
+
+6
+h
+J
+5
+4
+3
+2
+2
+3
+4
+5
+6
+8
+M? (GeV2)0.4
+(GeV)
+0.3
+0.2
+0.1
+0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+M? (GeV?)6
+schicker˙diff2022
+printed on January 31, 2023
+6. The final-state resonance mass distribution
+The (π, b)-trajectory consists of π and b-resonances with quantum num-
+bers (P,C)=(−, +), and (P,C)=(+, −), respectively. The final state shown
+in Fig. 3 can hence contain two π-resonances, or two b-resonances.
+PC=(-,+)
+M2
+1 ( ˜
+Sπ
+1 )
+M2
+2 ( ˜
+Sπ
+2 )
+arb. units
+PC=(+,-)
+M2
+1 ( ˜
+Sb
+1)
+M2
+2 ( ˜
+Sb
+2)
+arb. units
+Fig. 5: Two-dimensional mass distribution of the final-state resonances.
+In Fig. 5, the two-dimensional distribution of squared masses is shown
+for ˜s = 9 GeV2 for the case of two π-resonances on the left, and the corre-
+sponding distribution for two b-resonances on the right. Here, ˜s denotes the
+center-of-mass energy of the two initial-state Pomerons as shown in Fig. 3.
+7. Acknowledgements
+This work is supported by the German Federal Ministry of Education
+and Research under reference 05P21VHCA1. An EMMI visiting Professor-
+ship at the University of Heidelberg is gratefully acknowledged by L.J.
+REFERENCES
+[1] A.I.Bugrji et al.
+Dual Amplitudes with Mandelstam Analyticity.
+Fortschr.
+Phys. 21, 427, 1973.
+[2] F.Reidt. Analysis of Double-Gap Events in Proton-Proton Collisions at √s =
+7 TeV with ALICE at the LHC. Master thesis, University Heidelberg, 2012.
+[3] R.Schicker R.Fiore, L.Jenkovszky. Resonance production in Pomeron-Pomeron
+collisions at the LHC. Eur.Phys.J.C 76, 1, 38, 2016.
+[4] R.Schicker R.Fiore, L.Jenkovszky. Exclusive diffractive resonance production
+in proton-proton collisions at high energies. Eur.Phys.J.C 78, 6, 468, 2018.
+[5] N.Isgur S.Godfrey. Mesons in a Relativized Quark Model with Chromodynam-
+ics. Phys.Rev.D 32, 189, 1985.
+[6] P.A.Zyla et al. Particle Data Group. Prog.Theor.Exp.Phys. 2020, 083C01.
+[7] E.Predazzi A.Degasperis. Dynamical Calculation of Regge Trajectories. Nuovo
+Cim. Vol.A 65, 764, 1970.
+
+240
+220
+200
+180
+160
+140
+120
+100
+80
+60
+40
+20
+0
+8
+7
+6
+5
+8
+4
+7
+6
+3
+5
+2
+4
+3
+L
+2
+1
+0
+0300
+250
+200
+150
+100
+50
+0
+8
+7
+6
+5
+8
+4
+7
+9
+3
+5
+2
+4
+3
+1
+2
+1
+0
+0

Q9FPT4oBgHgl3EQfpTVX/content/tmp_files/load_file.txt

@@ -0,0 +1,165 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf,len=164
+page_content='Multiple pion pair production in a Regge-based model∗  Rainer Schicker  Physikalisches Institut, University Heidelberg, Heidelberg  in coll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' with Laszlo Jenkovszky  Bogolyubov ITP, National Academy of Sciences of Ukraine, Kiev  Received January 31, 2023  Central diffractive event topologies at the LHC energies can be identi-  fied by two different approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' First, the forward scattered protons can  be measured in Roman pots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Second, a veto on hadronic activity away  from midrapidity can be imposed to define a double-gap topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Such  a double-gap topology trigger has been implemented by the ALICE col-  laboration in Run 1 and Run 2 of the LHC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The analysis of these events  allows to determine the charged-particle multiplicity within the acceptance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  The excellent particle identification capabilities of ALICE allows to study  two-track events both in the pion and kaon sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Events with measured  charged particle multiplicity larger than two can arise from multiple pair  production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' A Regge-based approach for modeling such multiple pair pro-  duction is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Introduction  Double-Pomeron fusion at hadron colliders results in a double-gap event  topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Such a topology is defined by hadronic activity at or close to  midrapidity, and the absence thereof away from midrapidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The multi-  plicity distribution of such double-gap events has been measured in the  ALICE central barrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' To better understand such multiplicity distributions  we present here a Regge-based approach for multiple pion pair production  in double-Pomeron events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' This model is based on a Dual Amplitude with  Mandelstam Analyticity (DAMA) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' In this approach, the production of  multiple pairs can be modeled by including a Pomeron-Pomeron-Reggeon  and a triple-Pomeron coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The amplitude at Pomeron level within his  DAMA formulation is given, and the resulting mass distributions for double  pion and double b-resonance production are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  ∗ Presented at ”Diffraction and Low-x 2022”, Corigliano Calabro (Italy), September  24-30, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  (1)  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='13137v1  [hep-ph]  30 Jan 2023  2  schicker˙diff2022  printed on January 31, 2023  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Multiplicity distribution of double-gap events  The charged-particle multiplicity in the ALICE central barrel has been  analyzed in LHC Run 1 for both minimum bias and double-gap events [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  0  2  4  6  8  10  )  ch  (n  MinBias  N  )  ch  (n  DG  N  ) =  ch  (n  DG  R  6  10  5  10  4  10  3  10  2  10  Data  Tuned Pythia6  Tuned Phojet  Pythia6  Phojet  Phojet w/o CD  Phojet CD only  ch  N  0  2  4  6  8  10  Ratio MC to Data  1  10  1  10  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 1:  Double-gap probability in ALICE central barrel as function of  charged-particle multiplicity (Figure taken from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 1, the probability of being a double-gap event is shown as func-  tion of the charged-particle multiplicity Nch in the ALICE central barrel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  The ALICE data are shown in black circles, whereas the results from Monte  Carlo generators are shown in different colors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' These probabilities clearly  show a maximum at Nch=1 and Nch=2, demonstrating that double-Pomeron  events are dominated by very low multiplicities as compared to minimum  bias events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' As indicated in this figure, none of the tested generators shows  reasonable agreement with the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' This discrepancy between the ALICE  measured double-gap events and the prediction of the tested generators  motivates the development of a model which can be used to analyze unlike-  sign two-track events resulting from single resonance decays, as well as the  higher-multiplicity events stemming from the decays of multiple resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' A Regge model for double-Pomeron events  The model for Pomeron-Pomeron-induced events presented in the fol-  lowing is based on the DAMA approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Pomeron-induced single-resonance  production has been presented in our previous studies [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Here, we extend  this DAMA approach to the production of multiple resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  schicker˙diff2022  printed on January 31, 2023  3  1 amplitude  subdiagram  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 2: Amplitude at hadron level (left), and Pomeron subdiagram (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 2, the amplitude for Pomeron-induced single-resonance produc-  tion at hadron level is shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  The subdiagram on the right  represents the amplitude for Pomeron-Pomeron → resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The cross  section at hadron level is derived by convoluting the subdiagram cross sec-  tion with the Pomeron flux of the proton F P  prot(t, ξ) defined by  F P  prot(t, ξ) = 9β2  0  4π2 [F1(t)]2ξ1−2α(t),  (1)  with F1(t) the elastic form factor, and α(t) the Pomeron trajectory [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  In the DAMA approach, multiple-resonance production can be modeled  by introducing a Pomeron-Pomeron-Reggeon (PPR) coupling with subse-  quent splitting of the intermediate Reggeon into the two final-state Reggeons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  Alternatively, the same final state can be formed by a triple-Pomeron (PPP)  coupling with the intermediate Pomeron decaying into the two Reggeons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  1  αP  t1  αP  t2  g1  g2 ˜s  αR(˜s)  ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  ˜t  +  ˜S1(M2  1 )  ˜S2(M2  2 )  (˜s=t1+t2)  g3  g4  αP(˜s)  ˜S1(M2  1 )  ˜S2(M2  2 )  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 3: Subdiagram for PPR amplitude (left), and PPP amplitude (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  The DAMA amplitude for the subdiagram shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 3 is given by  00000000000000004  schicker˙diff2022  printed on January 31, 2023  APP→ ˜S1 ˜S2(˜s, ˜t, M2  1 , M2  2 )) =  1  �  M2  1 M2  2  �  PPR,PPP  �  n  gigjebα(˜t)  n − α(˜s) ,  (2)  with the first summation over the two amplitudes of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 3 defined by the  PPR coupling with gi,gj=g1,g2, and the PPP coupling with gi,gj=g3,g4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The  index n sums over the spins of the resonances of the intermediate trajectory  which connects the vertices i, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' From this amplitude, the cross section at  Pomeron level is derived by the optical theorem  σt(˜s, M2  1 , M2  2 ) = ℑm A(˜s, ˜t=0, M2  1 , M2  2 ),  (3)  with the imaginary part of A(˜s, ˜t, M2  1 , M2  2 ) defined by αR(˜s) and αP(˜s) for  the PPR and the PPP diagrams of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Reggeizing q¯q states in the light quark sector  The final-state mesons derive from the decay of the meson resonances  lying on the two Regge trajectories ˜S1 and ˜S2 as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' In or-  der to be able to include mesonic bound states of different radial and orbital  excitations, a unified description of q¯q bound states in the different flavour  sectors is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Such a unified description of q¯q bound states including a  confinement potential, a spin-orbit, a hyperfine and an annihilation inter-  action is presented in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The solutions for these q¯q bound states are  given in spectroscopic notation n 2S+1LJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  spectr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' notation n 2S+1LJ:  n radial quantum number  S spin  L orbital ang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' momentum  J total ang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' momentum  n2S+1LJ  mass  PDG  mass  width  Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' [5]  (PDG)  (PDG)  11S0  150  π  140  0  11P1  1220  b1  1230  142  11D2  1680  π2  1672  258  11F3  2030  ——  ——  ——  11G4  2330  ——  ——  ——  Table 1: Masses and widths in MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  In Table 1, masses are presented for the isovector channel in the light  quark sector for the radial ground state for S,P,D,F and G-wave, and are  compared to the values given by the Particle Data Group [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The S- and  D-wave bound states calculated in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' [5] are identified with the π and the  π2 states of mass 140 and 1672 MeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The P-wave solution  is associated to the known b1 state of mass 1230 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' No candidates for  the predicted F- and G-wave bound states have so far been experimentally  identified [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  schicker˙diff2022  printed on January 31, 2023  5  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Non-linear complex Regge trajectory  The small but existing non-linear dependence of the spin of a resonance  to its mass squared can be used to make a Regge trajectory α(M2) a complex  entity with real and imaginary parts being related by a dispersion relation [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  Here, the real part is defined by the value of the spin, and the imaginary  part is related to the decay width Γ by ℑmα(M2  R) = Γ(MR)α  ′ MR, with α  ′  denoting the derivative of the real part of the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' In a simple model,  the imaginary part is chosen as a sum of single threshold terms  ℑm α(s)=  �  n  cn(s−sn)1/2�s−sn  s  �|ℜe α(sn)|θ(s−sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  (4)  In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 4, the coefficients cn are fit parameters, and the parameters sn  represent kinematical thresholds of decay channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The (π, b)-trajectory  A Regge trajectory, called the (π, b)-trajectory hereafter, is defined by  the values of mass and width of the S, P and D-waves shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  ----------------  ----------------------  ----------------------------  b  b  b  π  b1  π2  b3  π4  b5  π  b1  π2  s0  s1  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 4: Real part (π, b)-trajectory on the left, width function Γ on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 4 on the left, the three data points of the π, b1 and π2 states are  shown by black points, and the non-linear fit by the blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' On the right,  the widths of the π, b1 and π2 states are shown by black points, and the fitted  width function Γ by the blue line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The thresholds s0 and s1 used in the fit of  Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 4 are shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The thresholds s0=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='176 GeV2 and s1=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='27 GeV2  are defined by the decays π2 →3π and b1 →K ¯Kπ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' This fit of  the (π, b)-trajectory predicts a b3 state with mass of 2090 MeV and width  of 321 MeV, a π4 state with mass of 2437 MeV and width of 352 MeV, and  a b5 state with mass of 2738 MeV and width of 371 MeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  6  h  J  5  4  3  2  2  3  4  5  6  8  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' (GeV2)0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='4  (GeV)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='3  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='2  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='1  0  0  1  2  3  4  5  6  7  8  M?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' (GeV?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' )6  schicker˙diff2022  printed on January 31, 2023  6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The final-state resonance mass distribution  The (π, b)-trajectory consists of π and b-resonances with quantum num-  bers (P,C)=(−, +), and (P,C)=(+, −), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' The final state shown  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 3 can hence contain two π-resonances, or two b-resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  PC=(-,+)  M2  1 ( ˜  Sπ  1 )  M2  2 ( ˜  Sπ  2 )  arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' units  PC=(+,-)  M2  1 ( ˜  Sb  1)  M2  2 ( ˜  Sb  2)  arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' units  Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 5: Two-dimensional mass distribution of the final-state resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 5, the two-dimensional distribution of squared masses is shown  for ˜s = 9 GeV2 for the case of two π-resonances on the left, and the corre-  sponding distribution for two b-resonances on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Here, ˜s denotes the  center-of-mass energy of the two initial-state Pomerons as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='  7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' Acknowledgements  This work is supported by the German Federal Ministry of Education  and Research under reference 05P21VHCA1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' An EMMI visiting Professor-  ship at the University of Heidelberg is gratefully acknowledged by L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
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+page_content='Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
+page_content=' 2020, 083C01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/Q9FPT4oBgHgl3EQfpTVX/content/2301.13137v1.pdf'}
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+Translating Text Synopses to Video Storyboards
+Xu Gu1*, Yuchong Sun1*, Feiyue Ni1, Shizhe Chen2, Ruihua Song1†, Boyuan Li1, Xiang Cao3
+1Renmin University of China, Beijing, China,
+2Inria, ´Ecole normale sup´erieure, CNRS, PSL Research University,
+3Bilibili Corporation, Shanghai, China
+https://ruc-aimind.github.io/projects/TeViS/
+Abstract
+A storyboard is a roadmap for video creation which
+consists of shot-by-shot images to visualize key plots in a
+text synopsis. Creating video storyboards however remains
+challenging which not only requires association between
+high-level texts and images, but also demands for long-term
+reasoning to make transitions smooth across shots. In this
+paper, we propose a new task called Text synopsis to Video
+Storyboard (TeViS) which aims to retrieve an ordered se-
+quence of images to visualize the text synopsis. We con-
+struct a MovieNet-TeViS benchmark based on the public
+MovieNet dataset [15]. It contains 10K text synopses each
+paired with keyframes that are manually selected from cor-
+responding movies by considering both relevance and cine-
+matic coherence. We also present an encoder-decoder base-
+line for the task. The model uses a pretrained vision-and-
+language model to improve high-level text-image matching.
+To improve coherence in long-term shots, we further pro-
+pose to pre-train the decoder on large-scale movie frames
+without text. Experimental results demonstrate that our pro-
+posed model significantly outperforms other models to cre-
+ate text-relevant and coherent storyboards. Nevertheless,
+there is still a large gap compared to human performance
+suggesting room for promising future work.
+1. Introduction
+With the prevalence of video sharing platforms, more
+and more video creators are emerging with enthusiasm to
+create videos using their own text synopses. An initial and
+critical step in professional video creation is to translate a
+text synopsis into a video storyboard, which is a sequence
+of shot-by-shot images to visualize key plots in a screen-
+play. Creating a high-quality video storyboard is however
+challenging for amateurs. It not only requires one to put
+*These authors contributed equally to this work.
+†Corresponding author.
+relevant scenes, characters and actions in the video, but also
+demands for cinematic organizations of keyframes such as
+coherent transitions across shots etc. Hence, there are high
+application needs in assisting amateurs to create more pro-
+fessional video storyboards from their text synopses.
+Although existing works have made great progress in
+text-to-image retrieval [8,10,18,19,25,45], text-to-video re-
+trieval [2,22,23,44] and even text-to-video generation [14,
+35,42], they are limited in creating storyboards from texts.
+The text-to-image retrieval works can only produce static
+images without considering the dynamics of shots in the
+video. Text-to-video works are able to retrieve or gener-
+ate videos. Yet, most of them focus on short-term video
+clips with only a few seconds as shown in Fig. 1a. The
+images in these videos are highly redundant and cannot
+satisfy the requirement of a video storyboard for coherent
+keyframes [1, 32]. The visual storytelling works [7, 16, 30]
+are proposed to visualize text with a sequence of images, but
+they care more about the text-image relevancy while omit-
+ting long-term reasoning to make transition smooth across
+keyframes (see Fig. 1b). Moreover, the query texts in exist-
+ing works are visually concrete and descriptive, making the
+models less generalizable to more abstract and high-level
+text synopses such as the synopsis in Fig. 1c.
+In order to reduce the gap between existing tasks and
+realistic needs for storyboard creation, in this work, we pro-
+pose a new task called Text synopsis to Video Storyboard
+(TeViS). In the TeViS task, we aim to retrieve an ordered se-
+quence of images from large-scale movie database as video
+storyboard to visualize an input text synopsis. For this pur-
+pose, we collect the MovieNet-TeViS benchmark based on
+the public MovieNet dataset [15]. The MovieNet dataset
+contains high-level text synopses for movies and a coarse-
+grained alignment between movie segments and text syn-
+opses paragraphs. We ask annotators to split paragraphs
+into semantically compact sentences and select a minimum
+set of keyframes from its aligned movie segment for each
+text synopsis sentence. Annotators should consider both
+relevancy to the text and cinematic coherency across frames
+1
+arXiv:2301.00135v1  [cs.CV]  31 Dec 2022
+
+Text-to-Video
+She looks back once more at SOMEONE then makes up her mind and starts running towards the exit.
+Highly redundant frames
+(a) Example from the LSMDC dataset [32] with detailed text descriptions and short-term video clips.
+Story-to-Image
+Incoherent transitions across images
+The dog was ready to go. He had a great time on the hike. And was very happy to be in the field. His 
+mom was so proud of him. It was a beautiful day for him.
+(b) Example from the VIST dataset [16] with story descriptions and incoherent image sequences.
+In London, Pamela settles back into her home, the one she can no longer afford.
+Text-to-Storyboard
+Cinematic coherent transitions across keyframes
+(c) Our constructed MovieNet-TeViS dataset for video storyboard creation from text synopsis.
+Figure 1. Comparison of our proposed Text Synopsis to Video Storyboard task and existing tasks.
+for keyframe selection.
+Finally, we obtain 10K text synopses and each paired
+with 4.6 keyframes on average.
+There are two unique challenges posed in our TeViS task.
+First, the text synopses are diverse covering a wide range of
+topics and some of them are also high-level and abstract,
+e.g., 2.76 concreteness score on average (vs. 2.99 in other
+video-text datasets such as LSMDC [32] and MAD [36]).
+Therefore, it is much more difficult to visualize texts with
+relevant images.
+Second, our text synopses correspond to video story-
+boards with much longer time range, e.g., 64 seconds on av-
+erage (vs. 4 seconds in LSMDC or MAD). A model should
+equip with long-term reasoning ability to ensure the image
+sequences are coherent in both event level and cinematic
+language level.
+We present an encoder-decoder framework as a start
+point to overcome the above challenges for video sto-
+ryboard generation.
+To improve the understanding of
+high-level text synopses, we finetune a pre-trained vision-
+language model (e.g., CLIP [25]) to retrieve relevant
+keyframes. Then an encoder-decoder framework with trans-
+former architectures is proposed to auto-regressively pre-
+dict image features, which can be used to retrieve and order
+images. However, it is difficult to train the decoder on a
+small dataset to retrieve coherent images. Inspired by the
+self-supervised pre-training paradigm in language model-
+ing [4, 11], we consider the cinematic coherency is a vi-
+sual language that is also possible to learn from large-scale
+unlabeled movie datasets. Therefore, we further propose a
+coherence-aware pre-training method that leverages large-
+scale movies without text synopses to pre-train the decoder.
+We design two settings to evaluate methods: i) an order-
+ing setting that provides models with oracle keyframes to
+re-order conditioning on the text, and ii) a retrieving-and-
+ordering setting that requires models to retrieve relevant
+frames from 500 candidate images and order them. Exper-
+imental results show that coherence-aware pre-training on
+unlabeled movies significantly improves the ordering per-
+formance . The larger the pretraining dataset, the better per-
+formance we could obtain in the downstream task. In addi-
+tion, the time interval in pre-training videos matters. Both
+2
+
+-Table 1. Comparison between MovieNet-TeViS and other movie datasets.
+Datasets
+avgDuration
+avg#Words
+SecondsperWord
+#unique bi-grams
+avgConcreteness
+LSMDC [32]
+4.1s
+9.03
+0.4539
+44.0K
+2.993
+MAD [36]
+4.04s
+12.69
+0.3188
+59.0K
+2.991
+CMD [1]
+132s
+18
+7.33
+83.2K
+2.598
+MovieNet-TeViS (Ours)
+63.7s
+24.82
+2.82
+134.5K
+2.761
+quantitative and qualitative results show that our model is
+able to create reasonable video storyboards. Nevertheless,
+there is still a long way to go to match the human perfor-
+mance on this challenging TeViS task.
+Our contributions are summarized as follows:
+• We propose the Text Synopsis to Video Storyboard
+task (TeViS) with the goal of retrieving an ordered se-
+quence of images to visualize high-level text synopsis.
+• We construct a MovieNet-TeViS benchmark based on
+MovieNet dataset [15]. It contains 10K text synopses
+with 4.6 keyframes on average for each synopsis.
+• We establish an encoder-decoder baseline and propose
+Coherence-Aware Pre-training on Movies to improve
+coherence in long-term video storyboards.
+2. Related Works
+Our work is related to previous works of two categories:
+text-to-vision and movie understanding.
+2.1. Text-to-Vision
+Text-to-vision aims to retrieve or generate visual infor-
+mation corresponding to an input text. Inspired by the suc-
+cess of the pre-training paradigm in NLP [4, 11], recent
+advances in text-to-image retrieval also leverage massive
+image-text pairs to pre-train a large model for retrieval [10,
+17,19,25,45]. These methods achieve promising results on
+caption-based image retrieval tasks such as MSCOCO [9].
+CLIP [22] adopts a dual-encoder architecture, uses 400
+million image-text pairs for pre-training with a contrastive
+loss, and shows strong generalization power on cross-modal
+alignment. Some works also pre-train video-language mod-
+els on large-scale video-text pairs [2, 23, 44].
+However,
+text-to-image technologies can only produce static images
+which can not describe the dynamics in text synopsis, while
+text-to-video retrieval target searching existing video clips,
+rather than picking up keyframes from clips to collage out
+something new, which is demanded for if users input new
+texts. Text-to-vision generation also develops rapidly. Ear-
+lier methods widely adopt GAN-based methods conditioned
+on text [24, 31, 46].
+Recent deep generative models use
+large Transformer networks [12, 27, 42] or diffusion mod-
+els [26, 33] that can generate high-quality images. Text-to-
+video generation has been explored recently by extending
+advanced text-to-image generation methods [14,35]. How-
+ever, even advanced text-to-video generation methods can
+only generate GIF-like short videos without complicated
+motions and dynamics.
+2.2. Movie Understanding
+Existing works on movie understanding mainly explore
+content recognition and cinematic style analysis.
+Con-
+tent recognition includes action [3, 21], scene recogni-
+tion [6, 13, 29], and text-to-video retrieval task that fo-
+cuses on movie datasets [1, 32]. Some works aim to ana-
+lyze shot styles [28, 40], movie genre [34, 47] from a pro-
+fessional perspective. There are also some movie-related
+datasets [1,15,32]. LSMDC [32] dataset contains short clips
+paired with human-annotated captions. Condensed Movie
+Dataset (CMD) [1] consists of key scenes from the movie,
+each of which is accompanied by a high-level semantic de-
+scription of the scene. The MAD [36] dataset is based on
+the LSMDC [32] dataset.
+Our constructed dataset is built on top of MovieNet [15]
+dataset, which is a large collection of movies annotated with
+many kinds of tasks such as scene segmentation, cinematic
+style classification, story understanding and so on. We use a
+subset of MovieNet annotated with text synopses of scenes.
+We manually construct video storyboard for the text syn-
+opses.
+Tab. 1 presents the comparison of our dataset and re-
+lated movie datasets. It shows that the duration of movie
+clips corresponding to a description in LSMDC and MAD
+is only 4 seconds and the average number of words in a de-
+scription is only 9-12, which is much lower than ours. It
+is impossible to extract a meaningful storyboard from such
+short clips. Our MovieNet-TeViS and CMD give a synopsis
+or summary of 64-second or 132-second video segments re-
+spectively and thus we can expect such text is higher-level.
+Compared to CMD, our MovieNet-Tevis uses more words
+to describe video segments with half the duration of CMD.
+This indicates that our text synopses provide more details
+than those in CMD. As a start of such a challenging new
+task, our dataset is the most appropriate in duration of video
+clip and semantic level of text.
+3
+
+Dorothy Gale  is an orphaned teenager who lives with 
+her Auntie Em and Uncle Henry on a Kansas farm in the 
+early 1900s.
+[subtitle]: "Aunt Em! Aunt Em!"
+"Just listen to what Miss Gulch did to Toto--", "Dorothy, please. 
+We're counting.“…
+shot0004_0
+shot0004_1
+shot0004_2
+shot0005_0
+shot0005_1
+shot0005_2
+Figure 2. Annotating keyframes of a storyboard for a text synopsis
+3. MovieNet-TeViS Dataset
+Our goal is to assist amateur video makers to create video
+storyboards from text inputs. Since it is hard to obtain orig-
+inal video storyboards from professional video makers, we
+decided to select keyframes from released movies to recon-
+struct a succinct storyboard that a human user can use as a
+shooting plan. In terms of the text inputs, previous works
+have collected aligned script [49], caption [32], Descrip-
+tive Video Service (DVS) [38], book [48], or synopsis [37]
+to movies. However, books cannot be well-aligned with
+adapted movies; DVS is hard to obtain and thus limited
+in scale; wiki plots are too coarse, while scripts and cap-
+tions are too detailed to compose for most non-professional
+users.
+We consider synopses are the most appropriate
+source which mimic the texts written by users in real sce-
+nario and contain desired level of details. MovieNet [15]
+and CMD [1] consist of such kind of synopses for movies.
+Although we could use the CMD dataset to scale up but we
+can expect that the CMD dataset would generate twice long
+sequence of images in a storyboard, which is much difficult
+to model, because it uses less words to describe the video
+clip with twice duration of our dataset.
+In the following, we first describe the dataset annotation
+process in Sec. 3.1 and then present analysis on our dataset
+in Sec. 3.2.
+3.1. Data Annotation
+MovieNet provides 4,208 text synopsis paragraph and
+movie segment pairs. A paragraph consists of 8 sentences
+(113 words) on average and a segmentation contains 95
+A1
+B1
+A2
+A3
+B2
+B3
+Figure 3. Simplifying a storyboard by deleting redundant images
+shots.
+It might be too difficult to learn semantic association and
+long-term reasoning over such long sequences with large
+variance. Therefore, we first split a paragraph into sentences
+and then align each sentence with a minimum number of
+keyframes in the movie segment.
+Fig. 2 shows the annotation interface.
+We present a
+synopsis paragraph sentence by sentence and all the shots
+aligned with the paragraph in MovieNet. For each shot, we
+display three evenly spaced frames as well as correspond-
+ing subtitles below the shot to help annotators understand
+the images better. The annotator should first select the sen-
+tences to form a text synopsis, and then choose a minimum
+number of images to visualize the text. We construct de-
+tailed guidelines to assure the quality of each annotated sto-
+ryboard as follows:
+1. The number of keyframes should be less than 20. Split
+the sentence if the number of selected keyframes is
+more than 20, or filter the sentence out of the dataset;
+2. Do not select adjacent similar images, e.g., for the ex-
+ample in Fig. 2, shot0004 1 should be deleted given
+shot0004 0;
+3. The image must add value to express the synop-
+sis sentence in terms of relevancy or coherency,
+e.g., shot0005 2 cannot add any new value given
+shot0005 0;
+4. If there is a basic conversation with a cycle of repeated
+images, only keep one pattern to make the storyboard
+succinct by selecting the first images in the first cycle
+and the last image in the last cycle. For example, in
+Fig. 3, AiBi is a basic conversation pattern and it has
+been repeated for 3 times. We ask annotators to select
+A1 and B3 to compose a complete conversation.
+To ensure data quality, our data are annotated in three
+rounds. We hire 60 annotators in the first round to select
+keyframes that are necessary in relevancy or a vision lan-
+guage; in the second round the annotators further simplify
+or revise the storyboards by consistent rules; and six vol-
+unteer experts in the third round review and finalize the se-
+lected keyframes.
+4
+
+3.2. Dataset Analysis
+Dataset statistics.
+Our collected MovieNet-TeViS dataset
+uses 2,949 paragraph-segment pairs from MovieNet after
+filtering improper examples by annotators. We sort story-
+boards by the number of keyframes ascendingly and use the
+first 10,000 pairs of a synopsis sentence in English and a
+video storyboard, i.e., a sequence of keyframes as our final
+dataset. There are 45,584 keyframes in total. The number
+of keyframes in a storyboard ranges from 3 to 11 and about
+60% storyboards consist of 3 or 4 keyframes. The average
+number of words in a synopsis sentence is about 24.
+In addition, MovieNet-TeViS covers 19 diverse movie
+genres. More details are presented in the supplementary
+material.
+Concreteness measurement.
+The concreteness level of
+texts has a large influence on visualization difficulty.
+To systematically measure the concreteness of texts, we
+leverage a concreteness database introduced by Brysbaert et
+al [5] to calculate average concreteness of words in synop-
+sis and compare with text descriptions of other datasets, i.e.,
+LSMDC, MAD, and CMD and show the results in Tab. 1.
+To be specific, Brysbaert et al. [5] create a database that
+ask annotators to assign concreteness ratings from 1 to 5
+for 40 thousand English words. The average ratings can
+evaluate the degree of how concrete a concept denoted by
+a word is. The larger value means more concrete. For ex-
+ample, the concreteness rating of “banana” is 5 while that
+of “love” is 2.07.
+As shown in the Tab. 1, our dataset
+has 2.76 average concreteness ratings while LSMDC and
+MAD have 2.99.
+This means that the text synopsis in
+MovieNet-TeViS is more abstract or higher-level than de-
+scriptions in LSMDC and MAD. CMD has 2.60 concrete-
+ness score which is slightly lower than ours. This makes
+sense because CMD use 18 words on average to describe
+132-second video clips whereas our MovieNet-TeViS uses
+24 words to describe 64-second video segments. As the first
+trial of a new task, our dataset has appropriate concreteness.
+Diversity measurement. Following [41], we use the num-
+ber of words, the number of unique n-grams and the num-
+ber of words with different POS tags to compare diversity
+of text description or synopsis in LSMDC, MAD, CMD
+and our dataset. For fair comparison, we randomly sample
+10,000 texts from LSMDC, MAD, and CMD datasets. We
+find that our built dataset MovieNet-TeViS has the richest n-
+grams, nouns, verbs, adjectives and adverbs. Due to space
+limitation, we only show the number of words and the num-
+ber of unique bi-grams results in Tab. 1. We present the full
+comparison in the supplementary material. From the Tab. 1,
+we observe that CMD is also richer than LSMDC. This
+supports our observation that LSMDC has caption based
+description whereas CMD and MovieNet have high-level
+summary of movie clips or segments. Our dataset is richer
+than CMD, which is consistent with what the seconds per
+word shows. When looking into our dataset, we find that
+many text synopses contain dialogues, psychological de-
+scriptions, shot languages, etc. Such free text styles are
+closer to that of our target non-professional video makers.
+4. Text Synopsis to Video Storyboard Task
+The Text Synopsis to Video Storyboard (TeViS) task
+aims to retrieve a set of keyframes and order them to vi-
+sualize the text synopsis. Assume we have a text synopsis
+T = {w1, w2, ..., wn} with n words, the goal of TeViS task
+is to retrieve m images from large candidate images and
+order them to visualize the text synopsis. The number of
+images m is different for each text synopsis T. We design
+two evaluation settings for the TeViS task: i) ordering the
+shuffled keyframes conditioned on the text, and ii) the task
+of retrieving then ordering.
+4.1. Ordering the Shuffled Keyframes
+Task.
+For a given text synopsis and its shuffled ground-
+truth images, how well can the models order them? This is
+a key step for creating a storyboard that needs to consider
+coherence across frames. To measure the long-term rea-
+soning capability of models for ordering, we let the models
+order the ground-truth images for this evaluation.
+Evaluation. For the ordering task, we are given a text syn-
+opsis and its shuffled ground-truth images, the models need
+to predict their order conditioned on text synopsis. We then
+can compute Kendall’s τ [20] metric to report the result.
+Kendall′s τ = 1 − 2 ∗ #Inversions
+N ∗ (N − 1)/2
+(1)
+where inversions are inverse-order pairs, i.e., the number of
+steps needed to switch to the original order. τ is always
+between -1 and 1, with 1 representing the full positive order
+and -1 representing the full inverse order.
+4.2. Retrieve-and-Ordering Keyframes
+Task.
+For a given text synopsis, how well can the models
+select the relevant images from a large set of candidates and
+then order them? This task is more practical in real situa-
+tions.
+Evaluation. For this evaluation, we are given a text synop-
+sis and a large set of candidate images. The candidate im-
+ages contain ground-truth images annotated by humans, and
+other negative images which are randomly sampled from
+other images in the corpus. The number of candidates in-
+cluding ground-truth and negative samples is 500. We con-
+sider both retrieval and ordering performance for this eval-
+uation, thus we use the product of Recall@K and Kendall’s
+τ as the final metric of this task. When some ground-truth
+5
+
+images cannot be returned at top K, the Kendall’s τ is calcu-
+lated upon the returned ground-truth images at top K only.
+5. Method
+To provide a start point for tackling the task, we pro-
+pose a text-to-image retrieval module based on a pre-trained
+image-text model (i.e., CLIP [25]), and an encoder-decoder
+module for ordering images.
+A coherence-aware pre-
+training method is further proposed to leverage large-scale
+movies to improve coherence across frames for the order-
+ing module. We also present several strong baselines built
+on top of CLIP for ordering.
+5.1. Text-to-Image Model for Retrieval
+We leverage a pre-trained image-text model CLIP to con-
+duct text-to-keyframe retrieval. During training, we ran-
+domly sample one frame from the ground-truth keyframe
+sequence to get positive image-text pair and frame from
+other sequences as negative for a text synopsis. Then we
+leverage a contrastive loss to maximize the similarity of
+matched images and texts while minimizing the similarity
+of unmatched images and texts, which is:
+Li2t = − 1
+B
+B
+�
+i=1
+log
+exp
+�
+I⊤
+i Ti/τ
+�
+�B
+j=1 exp
+�
+I⊤
+i Tj/τ
+�
+Lt2i = − 1
+B
+B
+�
+i=1
+log
+exp
+�
+T ⊤
+i Ii/τ
+�
+�B
+j=1 exp
+�
+T ⊤
+i Ij/τ
+�,
+(2)
+where Ii and Tj are the normalized embeddings of i-th im-
+age and j-th sentence in a batch of size B and τ is the tem-
+perature. The overall text-image alignment loss Lalign is
+the average of Li2t and Lt2i.
+5.2. Encoder-Decoder Model for Ordering
+Inspired by the encoder-decoder framework which is
+widely adopted in sequence generation tasks [9, 39, 43],
+we propose a Trans-TeViS model that adopts encoder-
+decoder architecture [39] to Translate Text synopsis to
+Video Storyboard. This design can not only sort the can-
+didate images but also handle the variable length prob-
+lem when creating video storyboards.
+As illustrated in
+Fig. 4, our model consists of a Transformer encoder ET
+for text encoding, and a Transformer decoder DT for image
+feature prediction. DT predicts the image features auto-
+regressively with a cross-attention mechanism to condition
+on text. These predicted features can be used to retrieve
+images by dot-product similarity. The model is optimized
+with an NCE loss in each prediction step with negative im-
+ages sampled randomly from a mini-batch:
+Ltrans = −
+1
+BM
+B
+�
+i=1
+M
+�
+m=1
+log
+exp
+�
+I⊤
+i,mIi,m/τ
+�
+�
+I′∈Ni,m∪Ii,m exp
+�
+I⊤
+i,mI
+′/τ
+�
+(3)
+where Ii,m is the normalized embeddings of the m-th im-
+age from the i-th image sequence from the batch, Ni,m is
+the normalized embeddings of the negative images sampled
+from the batch.
+Coherence-Aware Pre-training on Movies.
+Learning
+long-term reasoning for improving the coherence of re-
+ordered images is challenging, especially on a small dataset.
+It is hard for the model to learn sufficient movie-style evi-
+dence to make the transitions smooth across shots. Inspired
+by the success of the pre-training paradigm on NLP [4],
+which can learn language knowledge from massive data
+and then produce fluent sentences, we take a similar idea to
+leverage large-scale movies to learn the language of movies.
+Specifically, we pre-train the decoder part of our Trans-
+TeViS model with large-scale movie frame sequences with-
+out using text annotation. This method can be easily scaled
+up because movie frame sequences are easy to obtain.
+5.3. Additional Baselines for Ordering
+In addition to the proposed Trans-TeViS model, we de-
+sign three strong baselines based on CLIP for ordering as
+shown in Fig. 5:
+1) CLIP-Naive: we use CLIP to calculate the similar-
+ity between a text synopsis as query and its corresponding
+keyframes, and then order the keyframes based on the sim-
+ilarity scores.
+2) CLIP-Sliding: we first divide the sentences into sev-
+eral segments as a group of queries where the number
+of segments is equal to the number of its corresponding
+keyframes. We then use sliding window to use each seg-
+ment to retrieve the most similar keyframes in turn. Once
+a keyframe is chosen, this keyframe will be removed from
+the candidates.
+3) CLIP-Cumulative: we first divide the sentences into
+several segments as CLIP-Sliding. However, when doing
+retrieval, we accumulate each segment and retrieve the most
+similar keyframes, which considers more context. For ex-
+ample, to retrieve the second keyframe, we use the first two
+segments as the query. We also remove the keyframes from
+the candidates once they are chosen in previous step.
+6. Experiments
+We evaluate the performance of proposed methods on
+MovieNet-TeViS dataset for the text synopsis to video sto-
+ryboard task. We first describe the setup of experiment and
+then present the results of both ordering task and retrieve-
+and-ordering task. Finally, we show some qualitative re-
+6
+
+[START]
+Text Encoder
+Dorothy Gale is an orphaned
+teenager who
+lives with her
+Auntie Em and Uncle Henry on
+a Kansas farm in the early 1900s.
+Decoder
+Cross-Attention
+[END]
+…
+Retrieved Candidates
+𝐿𝑡𝑟𝑎𝑛𝑠
+Pre-Training
+Decoder
+Initialize
+Figure 4. The framework of our Trans-TeViS Model for TeViS task.
+𝒒𝟐
+text token
+𝒒𝟏
+𝒒𝟏
+𝒒𝟐
+𝒒𝟏
+𝒒𝟐
+𝒒𝟑
+𝒒𝟑
+𝒒𝟑
+𝒒
+𝒒
+image
+CLIP-Naive
+CLIP-
+Sliding
+CLIP-
+Cumulative
+Figure 5. Illustration of additional baseline models for ordering.
+sults.
+6.1. Experimental Setup
+Implementation Details. We utilize CLIP-ViT-B/32 as the
+backbone in all compared methods. The initial learning rate
+is set to 1e-6, and we use a linear learning rate scheduler to
+decay the learning rate linearly after a warm-up stage. The
+network is optimized by AdamW optimizer, with the weight
+decay value of 5e-2 and the batch size of 16.
+Pre-training datasets.
+We utilize the movies from
+CMD [1] dataset for pre-training the decoder of Trans-
+TeViS model. CMD dataset collects 7 to 11 clips with de-
+scriptions for each movie to cover the entire storyline. The
+CMD we exploited has 30K clips in the training set, 2K
+clips in the validation set, and 1K clips in the test. To bal-
+ance between information richness and computational com-
+plexity, we use a uniform frame sampling strategy to extract
+5 frames every clip for pre-training .
+Table 2.
+Results of ordering task.
+Our method Trans-TeViS
+achieves the best performance, though still leaving much room
+for improvement compared to human capabilities. [s − e] under
+Kendall’s τ denotes sequence length from s to e.
+Method
+Kendall’s τ↑
+all
+[3-5]
+[6-11]
+Human
+0.821
+0.860
+0.734
+CLIP-Naive
+0.183
+0.248
+0.036
+CLIP-Sliding
+0.230
+0.278
+0.123
+CLIP-Cumulative
+0.244
+0.291
+0.139
+Trans-TeViS
+0.261
+0.324
+0.120
+0.241
+0.246
+0.253
+0.261
+0.23
+0.235
+0.24
+0.245
+0.25
+0.255
+0.26
+0.265
+0
+10k
+20k
+30k
+Kendall’s τ 
+Number of clips from pre-training data
+Figure 6. Ablation study results of the pre-training dataset with
+different scales. The performance of our method improves as the
+scale of the pre-training data increases.
+6.2. Ordering Task
+Results.
+We conducted several experiments to verify the
+effect of different methods on the text synopsis to video sto-
+ryboard task, and results are shown in Tab. 2. The CLIP-
+Naive method achieves the poorest performance due to the
+lack of sequence modeling. The CLIP-Sliding and CLIP-
+7
+
+0.225
+0.261
+0.251
+0.243
+0.22
+0.225
+0.23
+0.235
+0.24
+0.245
+0.25
+0.255
+0.26
+0.265
+Kendall’s τ
+Average time interval between two frames
+6.15s
+12.32s
+18.41s
+26.80s
+14.16s
+Figure 7. Ablation study results of pre-training dataset with differ-
+ent average time interval between every two frames. Pre-training
+dataset with average time intervals similar to the MovieNet-TeViS
+(purple line of 14.16s) leads to a better performance.
+Cumulative methods outperform the CLIP-Naive method,
+proving to be more effective ways of applying CLIP, thanks
+to the ability to segment semantic information of the text
+and thus able to model sequences. The Trans-TeViS method
+achieves the best overall result of all the models, demon-
+strating the ability of sequence generation models to learn
+long-term information in keyframe sequences.
+In addition, a human study was conducted to make a bet-
+ter assessment of the task. We invited participants to reorder
+the shuffled keyframe sequences. The performance of hu-
+mans is presented in Tab. 2. Humans achieve much better
+performance than our best model, suggesting there is high
+potential for improvement.
+Analysis of Pre-training Data. To verify the impact of
+dataset scales in pre-training, we randomly sample subsets
+with different sizes from the CMD dataset for pre-training
+which has 30K clips originally.
+As shown in Fig.6, the method without any pre-training
+obtains the poorest performance, indicating the importance
+of Coherence-Aware Pre-training. It can be seen that the
+performance of the method improves as the scale of the pre-
+training dataset increases.
+Analysis of time interval. In the original CMD dataset, we
+use a uniform sampling strategy to extract 5 frames for each
+clip, and the average time interval (i.e. 1/fps) between every
+two frames is 26.8s, while in MovieNet-TeViS this number
+is 14.16s. To explore the impact of this gap, we extract
+frame sequences with the same sequence length and differ-
+ent time intervals from CMD as pre-training data. As shown
+in Fig.7, pre-training data with average time intervals simi-
+lar to the MovieNet-TeViS leads to a better performance.
+Table 3. Text-to-image retrieval performance on MovieNet-TeViS.
+We compare CLIP models with and without fine-tuning (ft).
+Method
+R@1↑
+R@5↑
+R@10↑
+R@50↑
+CLIP w/o ft
+5.73
+19.72
+28.98
+56.42
+CLIP
+7.48
+26.34
+38.94
+68.90
+Table 4. Results of Retrieve-and-Order Task. Our method Trans-
+TeViS outperforms other CLIP-based methods.
+Method
+Kendall’s τ↑
+R@10
+R@20
+R@30
+R@40
+R@50
+CLIP-Naive
+0.223
+0.169
+0.133
+0.143
+0.152
+CLIP-Sliding
+0.218
+0.204
+0.189
+0.208
+0.246
+CLIP-Cumulative
+0.226
+0.203
+0.188
+0.214
+0.249
+Trans-TeViS
+0.276
+0.271
+0.253
+0.251
+0.244
+6.3. Retrieve-and-Ordering Task
+Retrieval Results. We first evaluate the performance of
+text-to-image retrieval. We compare CLIP models [25] with
+and without fine-tuning on our MovieNet-TeViS dataset.
+As shown in Tab. 3, even without fine-tuning on our
+dataset, CLIP shows reasonable performance.
+The fine-
+tuned CLIP model can achieve much better performance.
+Retrieve-and-Ordering Results. We report the result of
+the Retrieve-and-Ordering Task in Tab. 4. It can be seen that
+the method of Trans-TeViS achieves the best performance
+while CLIP-Naive has the poorest, with CLIP-Sliding and
+CLIP-Cumulative in the middle. This result is consistent
+with the experimental result of the ordering task, suggesting
+that the difference in performance primarily stems from the
+difference in methods’ ability of ordering.
+6.4. Qualitative Results
+In addition to the quantitative results, we further carry
+out a case study on how well our proposed methods per-
+form in the TeViS task. As Fig. 8 shows, for the given
+text synopsis, our proposed Trans-TeViS method performs
+the best and can correctly order No.1, 3, and 4 keyframes
+and No.2, 3, and 4 ones. CLIP-Naive takes the synopsis as
+the whole to encode and thus it actually considers relevance
+only without any order information. It performs worst as ex-
+pected. Our proposed CLIP-Sliding and CLIP-Cumulative
+address the limitation because text synopsis is split into sev-
+eral text fragments and the ordering of keyframes depends
+on the ordering of text fragments. In this case, the text frag-
+ments are well aligned with ground-truth keyframes from
+human’s perspective, but it is still difficult for CLIP-Sliding
+and CLIP-Cumulative in ordering No.2 and 3. Our pro-
+8
+
+Suddenly enraged at the thought of Tracy, yet again, lying, cheating, seducing and manipulating her way into political 
+success for her own selfish reasons, Jim hurls a milkshake at the limousine, and then makes a quick getaway.
+Ground Truth
+Trans-
+TeViS
+CLIP-Navie
+CLIP-
+Sliding
+×
+×
+×
+×
+×
+×
+√
+×
+CLIP-
+Cumulative
+×
+×
+√
+×
+√
+×
+×
+√
+Figure 8. Qualitative examples of different models for the ordering task on our Movie-TeViS dataset.
+posed pre-training and transformer based model can cor-
+rectly order No.2 and 3, which shows the advantages in
+learning the visual language for storyboard creation.
+7. Conclusion
+In this paper, we introduce a novel TeViS task (Text
+synopsis to Video Storyboard), which aims to retrieve an
+ordered sequence of images to visualize the text synopsis.
+We also construct a MovieNet-TeViS dataset to support the
+task. To align the diverse text synopsis with keyframes,
+we utilize a pre-trained Image-Text model to overcome this
+challenge. We propose an encoder-decoder model called
+Trans-TeViS which translates text synopsis to keyframe se-
+quence.
+We also propose Coherence-Aware Pre-training
+on Movies to improve the long-term reasoning of the de-
+coder for ordering the keyframes. Ablation studies verify
+the effectiveness of our proposed model. Both quantitative
+and qualitative results show our method is better than other
+baselines.
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+Decoding Structure-Spectrum Relationships 
+with Physically Organized Latent Spaces 
+ 
+Zhu Liang,1 Matthew R. Carbone,2 Wei Chen,1 Fanchen Meng,1 Eli Stavitski,3 Deyu Lu,1,* Mark S. 
+Hybertsen,1,* and Xiaohui Qu1,* 
+1Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, USA  
+2Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973, USA  
+3National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA 
+* [email protected], [email protected], [email protected] 
+ 
+KEYWORDS 
+structure-spectrum relationship, interpretable machine learning, interpretable latent space, X-ray 
+absorption spectroscopy, X-ray absorption near-edge structure, autoencoder, deep learning   
+ABSTRACT 
+A new semi-supervised machine learning method for the discovery of structure-spectrum 
+relationships is developed and demonstrated using the specific example of interpreting X-ray 
+absorption near-edge structure (XANES) spectra. This method constructs a one-to-one mapping 
+between individual structure descriptors and spectral trends. Specifically, an adversarial 
+autoencoder is augmented with a novel “rank constraint” (RankAAE). The RankAAE 
+methodology produces a continuous and interpretable latent space, where each dimension can track 
+an individual structure descriptor. As a part of this process, the model provides a robust and 
+quantitative measure of the structure-spectrum relationship by decoupling intertwined spectral 
+contributions from multiple structural characteristics. This makes it ideal for spectral interpretation 
+and the discovery of new descriptors. The capability of this procedure is showcased by considering 
+five local structure descriptors and a database of over fifty thousand simulated XANES spectra 
+across eight first-row transition metal oxide families. The resulting structure-spectrum 
+relationships not only reproduce known trends in the literature, but also reveal unintuitive ones 
+that are visually indiscernible in large data sets. The results suggest that the RankAAE 
+methodology has great potential to assist researchers to interpret complex scientific data, test 
+physical hypotheses, and reveal new patterns that extend scientific insight. 
+I. INTRODUCTION 
+Structure-property relationships in materials encode fundamental physical knowledge and 
+enable a constructive design process to meet functional goals and guide materials discovery.[1–3] 
+Considering materials beyond the simplest of molecules or crystals, the full description of the 
+atomic-scale structure generally involves many separate quantities. Similarly, a collection of 
+properties can also be a complex data set. Consequently, the discovery and representation of 
+structure-property relationships pose significant challenges in the raw form of a “many-to-many” 
+map. Traditionally, expert intuition has been used to identify a few simple structure descriptors 
+
+2 
+ 
+that can be related to specific trends in properties through physical arguments and experimental 
+probes.[4] The emergence of high-throughput experiments and expanding databases of computed 
+materials properties invites the application of new, data-driven approaches to discover deeper, 
+more comprehensive structure-property relationships.[1–3] Significant progress has been made in 
+applying machine learning methods to physical datasets in recent years.[1–3,5–7] However, the 
+development of interpretable models,[8–12] highly desirable for use by physical science domain 
+experts, remains an ongoing challenge in the field. 
+Mapping the trends in a complex dataset, such as a collection of physical spectra measured for 
+a set of materials, to a reduced dimensional space is a common and productive first step. For 
+example, an autoencoder[13,14] (AE) maps each input spectrum to a point in a latent space of low 
+dimensionality while simultaneously training a decoder to perform the inverse mapping of a point 
+in the latent space back to a spectrum. However, the latent space variables do not inherently have 
+a physical interpretation.[9,10,15–19] Specifically for the structure-property relationship problem, the 
+data-driven discovery of correlative structure descriptors remains largely unsolved. 
+X-ray absorption spectroscopy (XAS) illustrates the complexity of structure-spectrum 
+relationships. XAS is a widely used technique for materials characterization[20,21] due to its element 
+specificity and its sensitivity to the local chemical environment of the absorbing atom.[22] 
+Furthermore, modern synchrotron facilities enable high-throughput experimentation and in situ 
+materials discovery methodologies.[23] In XAS experiments, a core electron is excited by an 
+incident photon to empty states. The X-ray absorption spectrum exhibits steps, called edges, at 
+clearly distinguishable energies where the absorption rises sharply. These are classified into K-, 
+L-, and M-edges corresponding to n=1, 2, and 3, where n is the principal quantum number of the 
+core electron. For a given absorbing element, the XAS spectrum is typically divided into two 
+regions relative to the edge. The region extending up to roughly 50 eV above the edge is referred 
+to as X-ray Absorption Near Edge Structure (XANES), and beyond that is the Extended X-ray 
+Absorption Fine Structure (EXAFS).[20] 
+XANES and EXAFS encode different types of information. EXAFS contains information 
+about the radial distribution of first shell neighbors, and potentially second and third shell 
+neighbors if the single-to-noise ratio is high enough. The analysis of EXAFS is well developed, 
+with robust approaches available for extracting structural information.[24,25] In the framework of 
+multiple scattering theory[26] (see Figure 1a), the EXAFS signal is dominated by the sum of direct 
+back-scattering terms. In contrast, XANES depends on the interference of a more diverse set of 
+interactions, such as the three-body path illustrated in Figure 1a. Furthermore, it contains rich 
+information about the electronic structure, specifically dipole-allowed transitions to low-lying 
+empty states. As a result, XANES spectral features can be correlated with local physical and 
+structure descriptors, such as oxidation state, coordination number, and local symmetry.[20,21,26,27] 
+However, XANES is more difficult to analyze than EXAFS. An in-depth analysis typically 
+requires details of the electronic states of the material as well as an accurate physical model of the 
+many-body core-hole effects.[26–29] Because of this inherent complexity, it is challenging to unravel 
+the structure-spectrum relationships in the analysis of XANES.  
+Structure-spectrum relationships enable the extraction of structural information from XANES 
+spectra measured on new materials, a core application of X-ray measurement in science and 
+engineering.  Consequently, the development of methods to infer structure descriptors from spectra 
+is a central research problem in XANES analysis.[21,30–35] In contrast to the well-posed problem of 
+
+3 
+ 
+computing a spectrum from a given material structure, this is an “inverse problem.” The inverse 
+problem may not always have a unique solution. Multiple structures may be consistent with a 
+measured spectrum. Solving the inverse problem relies on identifying a set of robust structure 
+descriptors that correlate to distinct spectral features or fingerprints. The search for such 
+relationships has inspired decades of research in X-ray spectroscopy. In 1920, Bergengren 
+discovered that the edge position strongly correlates with the oxidation state (OS) of the absorbing 
+atom (see Figure 1b).[30] Since then, OS has been a common focus of the K-edge XANES 
+analysis.[21,32,33,36] Other important structure descriptors include local symmetry (e.g., tetrahedral 
+versus octahedral) and coordination number (CN) of the cation, which are correlated with the 
+position and intensity of the pre-edge peak in early 3d transition metal oxides (Figure 1c).[21,31,35,37] 
+However, this empirical relationship shows strong system dependence and does not apply to late 
+3d transition metals with vanishing pre-edge features, as illustrated in Figure 1d.[37–39] Beyond the 
+first coordination shell, it is significantly more difficult to identify useful structure descriptors. 
+ 
+Figure 1. Real space scattering picture of X-ray absorption and exemplary spectral trends in K-edge 
+XANES. (a) The final state electron amplitude on the central atom includes the sum of scattering 
+contributions from all neighboring atoms. Scattering processes include: SS1 - single scattering from 
+the first coordination shell, SS2 - single scattering from the second coordination shell, and MS – an 
+exemplary multiple scattering path. (b) Simulated Mn K-edge XANES for a series of crystals in 
+which the oxidation state of Mn varies showing the shift of the main edge (gray arrow).  (c) 
+Simulated Ti K-edge XANES for three oxide crystals with varying titanium coordination numbers 
+showing the change in pre-edge feature intensity (gray arrow). CN4, CN5, and CN6 designate 4, 5, 
+and 6-coordinated motifs. (d) Same for Fe K-edge XANES but showing no simple trend. 
+SS1
+SS2
+MS
+(a)
+(b)
+(c)
+(d)
+
+4 
+ 
+Despite the utility of existing structure descriptors (e.g., OS and CN), there is a strong demand 
+to discover or engineer a sufficiently complete set of structure descriptors to support local structure 
+prediction in complex materials spaces. The significance of a structure descriptor strongly depends 
+on the underlying materials and the physical properties of interest. For example, the pre-peak 
+features in transition metal K-edge XANES are sensitive to the distortion of the cation octahedral 
+cage caused by substrate[40] or pressure.[41] Based on studies of Ti K-edge XANES and Li K-edge 
+electron energy loss spectra, local distortions are identified as structure descriptors that play an 
+important role in understanding the phase transformation and the fast lithium ion transport in 
+lithium titanate.[42–44] To capture the spectral trends of nanostructures, bond-length-based structure 
+descriptors have been investigated, which include metal-metal bond lengths in metallic (Pd K-
+edge)[18] and bimetallic nanoclusters (Pd K-edge and Au L3-edge),[45] average Fe-O bond length in 
+Fe oxide clusters (Fe K-edge),[34] and Co-C and Co-O bond lengths in a Co single atom catalyst 
+complex (Co K-edge).[46] In addition, the chemical composition (e.g., hydrogen content) can also 
+serve as a good descriptor for Pd nanoparticle catalysts exposed to H2.[18] These examples illustrate 
+the material specificity of the problem. Despite extensive studies,[32,33,47,48] a robust and widely 
+applicable approach for identifying structure descriptors still does not exist. 
+Intuitively, one can label a spectral dataset with various structure descriptors and use visual 
+inspection to search for qualitative trends. This empirical approach can succeed when there is an 
+obvious trend, such as the correlation of edge shift to OS in Mn K-edge XANES (Figure 1b) or 
+the correlation of pre-edge intensity to CN in Ti K-edge XANES (Figure 1c). However, this 
+approach fails when the spectral trend is too complex to identify by visual inspection, e.g., the 
+trend of CN in Fe K-edge XANES (Figure 1d). Nonetheless, in the latter situation, there is great 
+potential to use data analytics tools to extract statistically meaningful trends.[31,33,38,48–52]  
+In this study, we focus on methods capable of learning a latent space that simultaneously 
+represents spectra and correlates with structure descriptors. The resulting map is an interpretable, 
+multi-dimensional structure-spectrum relationship. We identify key technical challenges that must 
+be met to achieve this goal and develop new data analytics tools to address them. Specifically, we 
+build on the adversarial autoencoder (AAE)[53] by adding a new rank constraint that drives each 
+latent space variable to track a target structure descriptor. Taken together, our rank-constrained 
+adversarial autoencoder (RankAAE) method captures spectral variations along latent space 
+dimensions and correlates them with physically interpretable structure descriptors. We show that 
+the method works across a multidimensional latent space incorporating a set of structure 
+descriptors in a single training procedure. 
+To demonstrate the RankAAE method, we apply it to transition metal cation K-edge XANES 
+from eight 3d transition metal oxide families (Ti-Cu). A large database of simulated spectra is 
+calculated for crystal structures drawn from the Materials Project database, which exhibits diverse 
+local cation chemical environments. This represents a broad coverage of materials compositions, 
+including binary, ternary and quaternary oxides. The database includes over fifty thousand 
+individual cation spectra. With this database, we use the RankAAE methodology to build structure-
+spectrum relationships and characterize the effectiveness of the method. We show how it can be 
+used by a domain expert to explore candidate structure descriptors and their corresponding spectral 
+trends. In particular, this method can be used to compare the performance of different combinations 
+of structure descriptors in multi-dimensional models for structure-spectrum relationships. For the 
+transition metal oxides under study, the RankAAE method enabled us to both recover historical 
+structure-spectrum relationships[21,30,31,35–37,54] and to reveal new ones hidden in the data. 
+
+5 
+ 
+We henceforth summarize the structure of our manuscript. Latent space methods are briefly 
+reviewed in Section II.  Section III presents the main results demonstrating the use of the RankAAE 
+method.  The results are discussed in Section IV, including the physical interpretation of the 
+emerging trends.  Conclusions appear in Section V. Technical details of the methodology are 
+described in Section VI. 
+II. LATENT SPACE METHODS 
+Dimensionality reduction techniques pertinent to the structure-spectrum problem are briefly 
+described here. In particular, we place our RankAAE method in the context of previous work. 
+ 
+Figure 2. Dimensionality reduction with correlation to structure descriptors. (a, b) Principal 
+component analysis of simulated Fe K-edge XANES spectra. Data points are colored according to 
+the indicated structure descriptors: (a) Fe oxidation state (OS); (b) Fe coordination number (CN). 
+(c) Schematic illustration of feature entanglement in the spectrum latent space {z1, z2} (left) and the 
+goal of transforming the latent space into {𝑧1
+′, 𝑧2
+′ } (right), where 𝑧1
+′ and 𝑧2
+′  align with structure 
+descriptors OS and CN, respectively. OS is represented by color (blue: low OS; red: high OS), and 
+CN is represented by shape (triangle: low CN; circle: high CN).   
+One example of a simple, linear method commonly used to extract trends in data is principal 
+component analysis (PCA). Liu et al. applied PCA to the Cu K-edge of CuxPdy bimetallic 
+nanoparticles and identified patterns correlated with CuxPdy cluster types.[55] Carbone et al. 
+performed PCA on K-edge XANES of eight 3d transition metal families and identified clear 
+patterns, where the data points in the reduced dimensional space are distributed into identifiable 
+clusters ordered according to the label of the absorbing cation CN=4, 5, and 6.[38] Similar patterns 
+are shown here for Fe OS and CN in Figure 2(a, b), where the simulated spectra were drawn from 
+the database used in the present study (see Methods). Fe OS exhibits a meaningful pattern in the 
+reduced-dimensional space, where the gradient of the OS is roughly along the horizontal direction. 
+Similarly, the ordered PCA pattern of Fe CN is roughly along the vertical direction, in stark 
+  
+  
+   
+   
+ 
+ 
+ 
+   
+  
+          
+   
+          
+   
+   
+   
+   
+          
+     
+     
+  
+ 
+  
+ 
+  
+
+6 
+ 
+contrast to the vague trend in the raw spectra of Figure 1d. The PCA decomposition of Fe K-edge 
+XANES spectra in Figure 2 provides strong evidence that OS and CN are good structure 
+descriptors. However, the variation of OS or CN is highly non-linear with respect to the PCA axes, 
+and the patterns of the two descriptors are thus intertwined, as illustrated schematically in Figure 
+2c (left). This example illustrates the limitations of standard linear dimensionality reduction 
+methods, such as PCA, for quantitative structure-spectrum mapping.    
+ 
+Figure 3. Network structure of the RankAAE method. A standard autoencoder (blue dashed 
+rectangle) creates a latent space representation of the data that faithfully reproduces the spectra in 
+the training dataset. An adversarial autoencoder (green dashed rectangle) introduces a discriminator 
+on top of the regular autoencoder that regularizes the latent space. RankAAE further adds a rank 
+constraint (orange) to the latent space to establish correlations between specific latent space 
+dimensions and target physical descriptors. 
+A more promising dimensionality reduction approach uses autoencoder (AE)-based 
+methods.[13,14] An AE generally consists of two neural networks trained in tandem to approximate 
+the identity function: an encoder that compresses vector data sets to a lower dimensional latent 
+space and a decoder that reverses this mapping (see the blue dashed box in Figure 3). An AE can 
+perform sophisticated non-linear data compression and capture statistically relevant information 
+in the latent space. For example, Routh et al. trained an autoencoder to transform simulated Pd K-
+edge XANES spectra of small Pd clusters to a latent representation and revealed a strong 
+correlation between structure descriptors (CN, interatomic distance, and hydrogen content) and 
+latent space variables.[18] However, in studies that require continuous sampling of the latent space, 
+the mapping created by a basic AE can be problematic. For example, when applied to image 
+reconstruction problems, some regions of the latent space do not decode to sensible images.[56] In 
+a chemical science example, the latent space of an AE trained on SMILES encodings of 
+molecules[57] exhibited “dead areas” that did not decode to valid molecules.[17] In the present work, 
+the spectrum reconstruction from the decoder can yield unphysical spectra (Figure S1a). These 
+failures are due to the lack of regularization of the latent space. Some points in the latent space 
+may not correspond to regions in the neighborhood of any previously seen training data, causing 
+the decoder to produce unpredictable results. Variational (VAEs),[10,17,58] Wasserstein (WAE)[9] 
+and adversarial autoencoders (AAEs, green dashed box in Figure 3)[53,59,60] tackle this problem by 
+regularizing the latent space during training, forcing the training data coverage of the latent space 
+to be more complete. Although different in technical detail, all result in models capable of 
+performing robust data generation (i.e., generative models). The latent space can then be sampled 
+               
+             
+       
+       
+      
+        
+        
+
+7 
+ 
+continuously, with decoded signals that remain valid in the target application, e.g., yielding 
+physical spectra. For example, a VAE has been used to analyze spectral functions,[19] and VAEs, 
+WAE and AAEs all have been used as generative models to search for new molecules/materials 
+from the latent space.[9,17,59,60] 
+Robust, data-driven methodologies to uncover structure-spectrum relationships and discover 
+new descriptors must address three key technical challenges:  
+1. Spectrum validity: a data-driven method is needed to construct a set of latent variables that 
+serves as a proxy for the spectrum. In practice, the latent space needs to be regularized to 
+ensure that all data points in the latent space correspond to physical spectra.  
+2. Structure mapping: the method needs to establish quantitative mappings between the latent 
+space and structure descriptors, such that the statistical importance of various descriptors 
+can be assessed and compared.  
+3. Feature disentanglement: the method must disentangle spectral contributions driven by 
+different underlying structural and chemical characteristics.  
+For example, in the case of XANES, the spectral variation reflects the net effects from multiple 
+sources that are often convoluted in overlapping energy ranges. Even with prior knowledge from 
+domain-specific expertise, it is currently impossible to disentangle the impact of multiple structure 
+descriptors. To achieve disentanglement, the latent space needs to be further organized, such that 
+each latent space dimension aligns with a specific structure descriptor. Currently, none of the off-
+the-shelf data analytics tools simultaneously satisfies these three requirements of spectrum validity, 
+structure mapping, and feature disentanglement.  
+New methods need to be developed to simultaneously address all three challenges. We focus 
+on VAE/AAE methods. Because they are generative models in nature, they already satisfy the 
+spectrum validity requirement. Joint training of a property prediction network with a VAE can 
+address the structure mapping requirement,[17] however it does not solve the feature 
+disentanglement problem. Specifically, even with the addition of joint training, each latent variable 
+does not represent a target property directly since it can be entangled with multiple physical 
+descriptors. To construct a direct latent variable-structure descriptor mapping, the latent space of 
+a VAE/AAE needs to be reorganized to align each dimension with a structure descriptor. Figure 
+2c (right) depicts this idea: the goal is to have the latent variable 𝑧1
+′ only depend on OS (symbol 
+color) monotonically, while 𝑧2
+′ only depends on CN (symbol shape) monotonically. To accomplish 
+this, additional constraints need to be engineered and applied to a VAE or AAE. 
+The RankAAE method, developed in this work, accomplishes this goal with a novel “rank” 
+constraint specifically designed to organize the latent space in the way depicted in Figure 2c. The 
+implementation balances the goal of aligning each latent space variable with a target structure 
+descriptor while minimizing the impact on the other, statistical characteristics of the latent space 
+representation learned with the AAE. The technical details are described in Methods. Applied to a 
+multi-dimensional latent space, the interplay of different structure descriptors is disentangled and 
+the RankAAE method associates each dimension of the latent space with a well-defined structure 
+descriptor.  
+III. RESULTS 
+For our database of transition metal oxides, we simulate the XANES spectra for each 
+symmetrically non-equivalent transition metal absorber using the multiple-scattering code, 
+
+8 
+ 
+FEFF9.[26,61] Details of the database development and computational methods are given in Methods.  
+Since XANES cation K-edge spectra probe local chemical environments, we investigate the 
+structure descriptors that encode them. For the transition metal oxides, the cation-oxygen network 
+characteristics are of particular importance (Figure 4a), although most crystal structures in the 
+database also incorporate counterions that influence key chemical and structural attributes of the 
+local transition metal and its environment. For simplicity, we restrict our attention to sites where 
+the nearest neighbor shell contains only oxygen. In addition to the two well-known descriptors 
+(OS and CN), we consider several other descriptors directed at capturing distortions in the nearest-
+neighbor shell of the absorbing cation and the influence of the second nearest-neighbor shell. The 
+considered descriptors are detailed in Methods, but we outline them here briefly. 
+1) The average of the coordination numbers for the nearest-neighbor oxygen atoms (OCN), 
+2) The second-nearest-neighbor coordination number (CN-2), 
+3) The spread in the cation-oxygen bond lengths expressed as the nearest-neighbor radial 
+standard deviation (NNRS), 
+4) The spread in the cation centered bond angles expressed as the nearest-neighbor angular 
+standard deviation (NNAS), 
+5) The minimum oxygen-oxygen distance on the edges of the nearest neighbor polyhedron 
+(MOOD), and 
+6) The point group symmetry order (PGSO). 
+ 
+Figure 4. Structure and XANES spectral descriptors. (a) Schematic local structure around the central 
+cation emphasizing the cation-oxygen network. Shaded areas are first coordination shells of cation 
+(light red triangle) and oxygen (light blue triangle). (b) Schematic spectral descriptors applicable to 
+typical cation K-edge XANES spectra in oxides: Eedge – edge position, Ipre – pre-peak intensity, Imain 
+– main peak intensity, Cpit – post edge curvature, Emain – main peak position, Epost – post edge 
+position, Ipost – post edge intensity. 
+To facilitate the discussion of structure-spectrum relationships, we adopt a set of spectral 
+descriptors that capture the main spectral characteristics seen in a typical transition metal oxide 
+XANES K-edge spectrum (Figure 4b). Described in more detail in Methods, these basic metrics 
+will be referenced throughout our work. 
+We present our results in two parts. First, we illustrate the use of the RankAAE method and 
+our validation of its performance for one family of materials, Vanadium (V) oxides. Then, we 
+illustrate the performance of the method across the full set of transition metal oxides considered in 
+this study. 
+Cation
+Anion
+(a)
+(b)
+Cpit
+Ipre
+Emain
+Imain
+Eedge
+Epost
+Ipost
+
+9 
+ 
+A. Application and Validation of RankAAE for Vanadium Oxides 
+The full scope of the V K-edge XANES data across the present vanadium oxide database is 
+shown in Figure 5a. To illustrate the structure-spectrum relationships in this raw data, the spectra 
+are colored according to the values of five structure descriptors in sequence from the top of the 
+figure: OS, CN, OCN, NNRS, and MOOD, respectively. Visual inspection of the examples in 
+Figure 5a shows color concentrations indicative of correlations between structure descriptors and 
+spectral features, as one expects on physical grounds. For example, the position of the edge and 
+the main peak (Eedge and Emain) shift with OS, in line with the trend observed in experimental 
+spectra.[21,31,35,37] Pre-edge peak intensity (Ipre) and main peak position and intensity (Emain and Imain) 
+change with CN, but Emain and Imain are also affected by OCN and MOOD. Higher energy features 
+above the main peak show similar characteristics. This illustrates the entanglement of the 
+contributions from different structure descriptors to trends in spectral features. This diverse dataset 
+exemplifies the challenges that domain experts face when trying to expand the scope of structure 
+descriptors in the analysis. While the overall trends in spectral features for OS and CN follow prior 
+domain experience and physical models, the trends for additional spectral descriptors are too 
+ambiguous or complicated to draw clear conclusions. A specific structure descriptor correlates 
+with spectral features in several locations across the spectrum and each of those features can be 
+affected by multiple structure descriptors. Taken together trends are obscured and physical 
+interpretation is non-trivial. 
+We illustrate the use of the RankAAE method (Figure 3, Methods) through application to this 
+vanadium oxide dataset. The RankAAE model is trained to map the spectra in the database to a 
+latent space (center of Figure 3) of chosen dimensionality 𝑁 through an autoencoding procedure. 
+For most of the results presented in this study we take 𝑁 = 6. As a special type of AAE, the latent 
+space resulting from the trained RankAAE model is regularized and each reconstruction maps to 
+a physical spectrum (Figure S1b). In contrast, a standard AE model can yield wider variance and 
+unphysical features in the spectra, such as excessive noise and even dramatic, sharp spikes (Figure 
+S1a). In addition, the rank constraint is enforced for selected latent space variables to align with 
+chosen structure descriptors, as described in Methods. Unconstrained dimensions represent 
+residual characteristics of the spectral data set beyond the constrained dimensions. Training is 
+repeated from scratch for each set of chosen structure descriptors. The details of dataset splitting 
+(training, validation, and test) are listed in Table S1.  
+The regularized and aligned latent space resulting from the RankAAE method allows us to 
+present spectral trends in an easily interpretable fashion. Reconstructed spectra are mapped from 
+the latent space with the trained decoder. Once the model is trained, each point in the latent space 
+decodes to a spectrum. In addition, due to our training procedure, each point should correlate to a 
+set of structure descriptor values according to the imposed constraints. If the method is successful, 
+the latent space will encode structure-spectrum relationships. By tracing a path through the latent 
+space, we sample that mapping, tracking the correlated structure descriptor values together with 
+the associated reconstructed spectra. In the simplest version of this picture, we can track the 
+evolution of spectra along each axis of the latent space while holding the other latent space values 
+constant, e.g., equal to zero (Figure S2a). With the correlation to target structure descriptors, this 
+can isolate specific spectral features that correlate to that structure descriptor. 
+Given the complexity of the dataset and the goal of quantifying the structure-spectrum 
+relationship in a multidimensional descriptor space, we want to sample the latent space more 
+
+10 
+ 
+systematically, not just along isolated, one-dimensional paths. To this end, for each latent variable 
+𝑧𝑖, we average over spectra corresponding to different values of {𝑧𝑗}𝑗≠𝑖. In this way, the evolution 
+of spectra along direction 𝑧𝑖 is statistically representative of the full dataset. Methods section 
+describes this averaging procedure. Figures S2a and S2b compare the sampling along each isolated 
+axis with the averaging procedure. The two approaches result in consistent trends. We adopt the 
+averaging procedure in this work for its statistical rigor.  
+ 
+Figure 5. Correlating structure descriptors to spectral trends for V K-edge XANES from vanadium oxides.  (a) 
+Simulated spectra of vanadium oxides colored according to the value of specific structure descriptors: OS, CN, 
+OCN, NNRS, MOOD, and no designation.  (b, c) Reconstructed spectra from trained RankAAE models. As 
+described in the text, each subplot maps the spectral trend associated with one of the six dimensions in the latent 
+space. Spectra are colored by the value of the corresponding latent variable. Five of the latent space variables are 
+constrained to correlate with target structure descriptors while the sixth variable is unconstrained. In (b), an initial 
+set of structure descriptors is chosen for training: OS, CN, CN-2, NNAS, and PGSO. In (c) a final set of structure 
+descriptors is chosen: OS, CN, OCN, NNRS, and MOOD. For each descriptor, the degree of correlation to the 
+corresponding latent space variable is quantified by an F1 score (for OS and CN) or the Spearman rank correlation 
+coefficient[62] (for the other descriptors), annotated next to each plot. The shaded area marks the location of the 
+primary spectral variation characteristic of the spectral trend. 
+   
+   
+   
+Energy (eV)
+Intensity (arb. units)
+Unconstrained
+Unconstrained
+OS
+CN
+OCN
+NNRS
+MOOD
+L
+H
+
+11 
+ 
+For the vanadium oxide dataset, Figures 5b and 5c show a series of trends for two trained 
+RankAAE models respectively, each with five constrained latent space dimensions and one 
+unconstrained dimension. Each subpanel shows the reconstructed spectral evolution along one 
+dimension of the latent space. The smooth variation in spectra highlights the success in 
+regularizing the latent space. The distinctive variations from panel to panel provide the input for 
+interpreting the trends in terms of the target structure descriptors and assessing their relative utility. 
+It is not known a priori which structure descriptors capture the most significant spectral 
+variation given a diverse dataset of materials structures. Figure 5b shows our “initial guess” for 
+these structure descriptors based on prior domain knowledge. We evaluate the results according to 
+several criteria. First, we monitor the quantitative correlation between the latent space variable and 
+the target descriptor as detailed below. Second, we look for emerging spectral trends in distinct 
+regions of the XANES spectrum. To be most useful for applications, those trends should be 
+distinguishable. Finally, we monitor the amplitude of the spectral variation captured by the 
+unconstrained latent space variable 𝑧6. In essence, the last metric captures the spectral variance 
+outside the scope of that driven by the target structure descriptors. Reducing it should indicate 
+improvements in the completeness of the descriptor set.  
+For the results in Figure 5b, we see that the target descriptors are captured with generally high 
+correlation (inset values in each panel). Furthermore, for each descriptor, a primary trend in the 
+spectra can be identified (shaded area). In addition, there are other spectral trends associated with 
+each descriptor across the energy range. For example, CN is correlated with both pre-edge intensity 
+and main peak intensity. Thus, while it is convenient to isolate localized spectral features for 
+discussion, each structure descriptor is associated with extended fingerprints with contributions 
+from spectral features across the full energy range. We also note an overlap in the trends. For 
+example, OS, CN, and NNAS all correlate to pre-edge intensity. Finally, the unconstrained 
+dimension exhibits a relatively small spectral variation. 
+The process of refining the choices of structure descriptors is iterative and still requires a 
+“human in the loop.” As an example, we retain the well-known OS and CN descriptors that clearly 
+show a strong, systematic spectral trend and consider alternatives for other structure descriptors. 
+Figure S3 shows the change in spectral trend as each of those structure descriptors is replaced one 
+by one. The OCN descriptor improves over CN-2 with a somewhat higher correlation to the latent 
+space variable and qualitatively supports a stronger spectral trend in the shape of the primary peak. 
+Introducing NNRS retains the correlation to the pit curvature while reducing the impact on pre-
+edge peak intensity. The MOOD descriptor exhibits a much-improved correlation to the latent 
+space variable compared to PSGO (0.86 versus 0.68). In this example, the spectral variation 
+attributable to the unconstrained latent space dimension remains relatively small. The spectral 
+trend for the final structure descriptor set is presented in Figure 5c. In comparing the trends with 
+the initial result in Figure 5b, the interaction between the descriptor choices can be seen. The pre-
+edge intensity trend for OS has been suppressed, and the spectral trend around the main peak shape 
+for CN has been concentrated.  Also, some of the overlaps in the spectral trends have been reduced.  
+In particular, the pre-edge peak intensity is now more specifically correlated to the CN descriptor, 
+and to a smaller degree to NNRS. This set is likely not unique, but it captures most of the spectral 
+variance in the vanadium oxide database.  
+A different perspective on descriptor development is to consider how the model evolves as 
+descriptors are added one by one. This is illustrated in Figure S4 for the five descriptors used in 
+
+12 
+ 
+Figure 5c. The quantitative correlation between each structure descriptor and the corresponding 
+latent space variable is stable. The qualitative spectral trends are also similar, although we observe 
+a clear sharpening of them. For example, the initial pre-edge feature in the OS trend systematically 
+reduces to essentially zero. The CN trend consolidates into clear pre-edge and main peak features.  
+At the same time, the OCN trend is capturing main peak shape changes. Finally, we also see the 
+stepwise reduction in the amplitude of the unconstrained latent variables as additional structure 
+descriptors are added. This strongly supports the notion that each new structure descriptor is 
+extending the model to capture additional variation in the spectra. It also illustrates how the 
+RankAAE approach overcomes the complexity of intertwined spectra trends in the raw spectral 
+data (Figure 5a). As the set of structure descriptors incorporated into the trained model is modified, 
+a set of distinguishable spectra trends emerges.  
+From the fully trained models, we can identify regions of the spectra with “primary spectral 
+trends,” each one linked to a specific latent variable 𝑧𝑖 and the associated structure descriptor 
+(Figures 5b and 5c).  Other significant concentrations of spectral variations constitute “secondary 
+spectral trends.” Together, they represent a spectral fingerprint. For the specific model for 
+structure-spectrum relationships shown in Figure 5c, we identify the following trends: 
+Oxidation state (OS): The edge position shifts to higher energy with the increase of atomic 
+charge due to shielding effects (Figure 5c, 𝑧1and Figure 4b, Eedge). In addition, there is a prominent 
+secondary spectral trend: the higher energy portion of the spectrum also shifts horizontally (Figure 
+4b, Epost). 
+Cation coordination number (CN): The pre-edge peak intensity decreases sharply with the 
+increase of CN (Figure 5c, 𝑧2 and Figure 4b, Ipre). In addition, the intensity change after the main 
+peak constitutes an identifiable secondary spectral trend (Figure 5b, Ipost). 
+Oxygen coordination number (OCN): The intensity of the main peak increases with the 
+increase of OCN (Figure 5c, 𝑧3 and Figure 4b, Imain). Overall, OCN caused smaller but more 
+focused changes than CN and alters the main peak shape. 
+Standard deviation in the nearest neighbor bond length (NNRS): The oscillation in absorption 
+intensities becomes weaker as NNRS increases, especially at higher energies (Figure 5c, 𝑧4 and 
+Figure 4b, Cpit). In addition, there is also a small variation in the pre-edge peak. In general, this is 
+a mild contribution to the total signal. 
+Minimum oxygen-oxygen distance (MOOD): This contribution to the total signal is even 
+smaller than the other descriptors. However, the variation at the main peak is strong: the position 
+of the main peak shifts to lower energy with increasing MOOD (Figure 5c, 𝑧5 and Figure 4b, Emain).  
+Unconstrained latent variable z6: By design, an AAE disentangles its latent space. In the case 
+of the RankAAE model presented here, 𝑧6  is disentangled from and contains information 
+supplementary to the constrained variables {z1, z2, … , z5}. In other words, the plot for 𝑧6 in Figure 
+5c represents all other spectral trends not captured by the five structure descriptors.  
+Having illustrated the characteristics of RankAAE models, we now describe quantitative 
+validation. First, we characterize the correlation between target descriptors and latent space 
+dimensions. To be clear, the value of a latent variable does not equal the value of the structure 
+descriptor directly. However, each latent variable is nearly monotonically correlated to a specific 
+structure descriptor, due to the soft constraint in our loss function. For the same model presented 
+in Figure 5c, latent space values are plotted against calculated structure descriptor values in Figure 
+
+13 
+ 
+6 for the test set in scatter plots for each of the five descriptors. The correlations are visually 
+apparent. They are quantified using the F1 score[63] for categorical variables (OS and CN) and the 
+Spearman rank correlation coefficient[62] (SRCC) for continuous variables (OCN, NNRS, and 
+MOOD). An F1 score ranges from 0 for poor classification performance to 1 for perfect 
+classification performance. An SRCC value of 0 occurs when there is no correlation; a value of 1 
+indicates a perfect, monotonic, positive correlation. Furthermore, the degree of correlation depends 
+on the choice of descriptor, as illustrated in Figure S3. As shown by the annotations in Figure 5c 
+and summarized in Figure S5a, all the F1 scores are above 0.91 and all the SRCCs are above 0.86, 
+highlighting the ability of the RankAAE methodology to drive latent variables to track structure 
+descriptors in a nearly monotonic fashion. 
+ 
+Figure 6. Correlation between the RankAAE latent space variable (x-axis), target structure 
+descriptor (y-axis), and emergent spectrum descriptor (color) for the vanadium oxide data set. 
+Violin plots are shown for categorical variables OS (a) and CN (b). Scatter plots are shown 
+for continuous variables OCN (c), NNRS (d), and MOOD (e).  
+Next, the emergent spectral trends encoded in the reconstructed spectra in the RankAAE model 
+in Figure 5c are compared to features in the ground truth spectra, i.e., the spectra directly from the 
+data set. For simplicity, we focus on the primary spectral trends that have been identified in Figure 
+5c. We compute these primary spectral descriptors for each of the underlying spectra in the test set 
+and present them in Figure 6. Visual inspection verifies that the evolution of the spectral 
+descriptors tracks both the latent variables and the structure descriptors. To quantify these 
+relationships, the F1 scores and SRCCs between the spectral descriptor in the data set and the latent 
+space variable are computed (Figure S5b). The coefficients are smaller than those for the structure 
+descriptors (Figure S5a). However, they are still significant, especially considering that only one 
+local spectra feature has been used. Also, the additional fluctuations associated with the spectral 
+features as emergent in a multi-dimensional space play a role. Finally, the reduction of the 
+
+14 
+ 
+correlation analysis to single dimensions is an oversimplification. For example, for z5, the 
+correlation with spectral descriptor (Emain) is only clear after categorizing the data points according 
+to the CN because the main peak variation is also affected by CN. This effect is shown in Figure 
+S6 using the test portion of the vanadium oxide dataset as an example. The data points aggregate 
+to a strip for each CN category. Inside each strip, the correlation between z5 and Emain is much 
+clearer, especially for CN5 and CN6, than when considering all the data in aggregate. This 
+illustrates that a local spectral feature, such as Emain, can still be affected by multiple structure 
+descriptors.  
+Taken together, these validation results demonstrate that the trained RankAAE model provides 
+multidimensional, physical structure-spectrum relationships for the vanadium oxide dataset.   
+B. RankAAE Performance across the Full Set of Transition Metal Oxides 
+The characteristics, trends, validation, and the final set of five structure descriptors shown for 
+the specific example of V oxides carry over to the full set of 3d transition metal families considered 
+in this work. The spectral data colored by the structure descriptors reveals trends with OS and CN, 
+but ambiguity remains when considering the larger set of descriptors (Figure S7). RankAAE 
+models are trained on the datasets of Ti, Cr, V, Mn, Fe, Co, Ni, and Cu oxides.  An independent 
+model is created for each material family. The same set of structure descriptors is used for the 
+entire series to facilitate comparisons. The reconstructed spectra versus each latent space 
+dimension exhibit smooth trends (Figure S8). The amplitude of spectral variation for the 
+unconstrained latent space variable z6 varies across the series. This suggests an opportunity to 
+further fine-tune the choice of structure descriptors. Examining the spectral trends, the same set of 
+spectral features can be tracked, although the specific energy ranges (relative to the edge) and the 
+relative importance of the primary versus secondary vary. 
+ 
+Figure 7. RankAAE derived spectrum variation trends for CN (a) and OCN (b) over Ti, V, Cr, Mn, 
+Fe, Co, Ni, and Cu oxides data sets. The spectra are colored by underlying latent variables z2 and z3 
+targeting CN and OCN, respectively.  
+                      
+   
+   
+                      
+           
+           
+
+15 
+ 
+To illustrate these trends further, Figure 7a shows the trend of the spectral variations for CN 
+across the series of first-row transition metals. The RankAAE models reveal two distinct primary 
+trends. First, like for V oxides, the pre-edge peak variation is the primary spectral trend for Ti, Cr, 
+and Mn oxides. The pre-edge peak intensity is strongest for V and Cr, intermediate for Ti, and 
+small, but discernable for Mn. Second, the late transition metals (Fe, Co, Ni, and Cu) exhibit 
+minimal intensity in the pre-edge peak region of the XANES K-edge spectrum.[37,38] For these 
+metals, the secondary spectral trend overweighs the primary trend: the second peak intensity above 
+the main edge, which is at least 5 eV away from the main peak (Figure 4b, Ipost), correlates with 
+CN. 
+ As a second example, the spectral trend for OCN is shown in Figure 7b. There is a universal 
+pattern: spectral intensity changes at the main peak. For Cu oxides, the intensity variation is 
+localized to that around the main peak. It is also dominant for Mn oxides, Co oxides and Ni oxides. 
+Other cases show more extensive, secondary spectral variation. Overall, the spectral features 
+associated with OCN are distinguishable from those driven by CN. However, Cr oxide is an 
+exception, where the trend associated with OCN is very similar to the trend for CN. 
+Validation of all the RankAAE models using the test sets for each material family shows strong 
+correlations between latent space values and the target structure descriptors as well as the identified 
+spectral features (Figure S9). Quantitatively the F1 scores and SRCCs for structure descriptor to 
+latent space values are high, as shown as insets in Figure S8 for each material and descriptor, as 
+well as in summary form in Figure S5a. Most F1 scores are above 0.90 with a minimum of 0.82. 
+Most SRCCs are above 0.85 with a minimum of 0.80. This indicates that the latent space in the 
+RankAAE models captures the changes in structure descriptors sufficiently across the entire set of 
+3d transition metal families. OS and CN exhibit high correlations in all cases. The correlation for 
+the three new structure descriptors varies across the metal series, but is very strong in general, 
+showing their potential for motif characterization in XANES analysis. Quantitative validation 
+against the primary spectral descriptors (Figure S5b) supports the qualitative results presented in 
+Figure S9. Some specific cases do highlight the limitations of relying on a single spectral descriptor. 
+For example, the low correlation (SRCC=0.13) for the Mn oxide CN descriptor with the pre-edge 
+intensity spectral descriptor traces to the relatively low intensity of the pre-edge feature, as noted 
+above. However, the SRCC increases to 0.44 if Ipost is used. Consistently, Ipost captures CN better 
+as a spectral descriptor for late transition metals (Mn-Cu). 
+IV. DISCUSSION 
+The RankAAE model described here creates a multidimensional, continuous spectral 
+representation that maps a target set of structure descriptors to XANES K-edge spectral 
+characteristics. The spectrum validity is a key prerequisite for its interpretability. A meaningful 
+structure-spectrum relationship can only be established on a properly regularized latent space.   
+Each point in the latent space must reconstruct to a physical XANES spectrum. As illustrated in 
+the vanadium K-edge data set in Figure S1a, a basic AE model does not meet the spectrum validity 
+criteria, as discussed in the Introduction. In fact, the AE model does reconstruct the spectra very 
+well for latent space points close to areas well represented in the original training data set, 
+evidenced by the dominance of “normal” spectra in Figure S1a. However, there are regions with 
+values that fall in the valid range overall, but for which the reconstruction can become 
+catastrophically poor. These latent space points fall where the model is interpolating in regions 
+sparsely represented in the training original dataset.[56] In contrast, the latent space of the 
+
+16 
+ 
+RankAAE model is regularized by the adversarial constraint, which enforces spectrum validity, as 
+shown in Figure S1b, consistently produces physically meaningful data points. 
+Analysis of the reconstructed spectra along specific dimensions in the latent space of the 
+RankAAE associated with each structure descriptor reveals well-defined spectral trends. For use 
+by domain experts, it is essential that these trends are robust, and that there is a reasonable 
+workflow to allow the expert to assess which choices of a structure descriptor are capturing the 
+essential spectral variations, and that the trends are physically interpretable. 
+The workflow for developing multidimensional models demonstrated here exhibits stability in 
+the spectral trends that emerge. We revisit the test sequence where we start with a single 
+constrained latent space variable and one free dimension. We also add a latent space variable that 
+is constrained to a pseudo-random number sequence as a structure descriptor (NOISE) to probe 
+the limit of the noise level in deriving spectral trends. Thus, we start with 𝑁 = 3, and add 
+dimensions together with constraints in succession. The root of this chain is OS, followed by CN, 
+OCN, NNRS, and finally MOOD. For the vanadium oxide dataset, following the same analysis as 
+presented in Figure 5c, the appropriately averaged, reconstructed spectra along each dimension are 
+shown in Figure S4. Several important points emerge. First, by visual inspection, the trends shown 
+for each dimension remain qualitatively consistent as new dimensions and structure descriptors 
+are added to the model. For example, for OS, the edge shift and horizontal translation remain stable 
+with different combinations of structure descriptors, although the amplitude of the changes varies 
+quantitatively. In particular, in the test using OS as the only structure descriptor, the pre-edge 
+variation is not localized. Instead, it appears in two latent variables (z1 and unconstrained z3). With 
+the addition of CN and more structure descriptors, it is localized to z2.  This reflects the resolution 
+of entangled trends as further structure descriptors are added. Second, the effectiveness of the 
+constraint in forcing the latent space variables to track the target descriptors is nearly independent 
+of the dimensionality of the model, as shown by the F1 scores and SRCCs that quantify the 
+correlation between latent space variables and target structure descriptors (Figure S4). Third, the 
+noise spectral trend always appears in a very narrow range independent of dimensionality and the 
+choices of constraint on other latent dimensions. Finally, the amplitude of the unconstrained 
+spectral trend systematically goes down as new dimensions and structure descriptors are added to 
+the model. This gives a qualitative guide indicating that each additional descriptor is capturing 
+new spectral variation. With the full complement of five structure descriptors in the model, the 
+spectral variation associated with the unconstrained variable is close to the noise level in vanadium 
+oxides. 
+Another important aspect of the workflow for a domain expert is to distinguish between 
+different choices of target descriptors. Several criteria have already been illustrated, including 
+distinguishable spectral trends, quantitative correlations between target structure descriptors and 
+constrained latent space variables, and the amplitude of the unguided spectral trend. Here we 
+further analyze the degree to which each new descriptor results in a qualitatively distinguishable 
+spectral trend. In order to capture the energy dependence of each trend when generating new 
+spectra by traversing the latent space, we examine a “differential spectrum”. We specifically 
+represent the total dynamic range of the trend by taking the difference between the last and first 
+spectrum corresponding to the 95th percentile and 5th percentile values in the range of the latent 
+space variable. We revisit the comparison of trends for different structure descriptors presented in 
+Figures 5 and S3 for vanadium oxides through the differential spectra shown in Figure S10. This 
+representation of the trends also exhibits the qualitative stability of trends for OS and CN as other 
+
+17 
+ 
+structure descriptors are changed. Further, as previously highlighted, the overall amplitude of the 
+unguided trend is reduced as the choice of structure descriptors is refined. As the other structure 
+descriptors are swapped out, the qualitative changes in the trends can be seen. For example, OCN 
+more narrowly isolates the change in the main peak intensity while CN-2 is convolved with the 
+shift of the main peak energy. Also, the trend for MOOD captures changes to the main peak shape, 
+position, and intensity as compared to the relative spread-out trend for PGSO. The spectral trends 
+are an emergent characteristic of the RankAAE models, so they are not constrained to be linearly 
+independent. Nonetheless, in the workflow, the degree of overlap can be monitored through the 
+intercorrelation measured by the Cosine similarity,[64] with the maximum values noted in each 
+panel of Figure S10. Overall, as the workflow proceeds, the degree of overlap is reduced. In our 
+final model, the CN descriptor and unconstrained dimension result in the most independent trends 
+while OS, OCN, NNRS, and MOOD derived trends remain correlated with each other to a 
+moderate degree by this measure. Finally, this analysis points in the direction of future 
+investigations to refine spectral fingerprints beyond the simplification of the specific, localized 
+spectral descriptors in Figure 4b. 
+Turning to physical interpretation, each of the primary and secondary spectral trends identified 
+above with structure descriptors can be understood. Among all the structure-spectrum relationships, 
+the correlation between OS and edge position is probably the most well-known.[30] Conceptually, 
+it corresponds to a basic physical principle: the energy cost to excite a core electron increases as 
+the cation becomes more positively charged. The secondary spectral trend, the correlated shift of 
+the post-edge intensity to higher energy, corresponds to the famous “molecular ruler”,[20] also 
+referred to as Natoli’s rule:[21] the local bond length is inversely linked to the position of a peak. 
+Here the ionic radius changes inversely with the atomic charge for the 3d transition metal cation, 
+driving the metal-oxygen bond length. These two empirical rules were proposed independently. 
+The unified spectral trend for OS that emerges from the RankAAE models captures these 
+correlated effects of the cation oxidation state.  
+The pre-edge peak is linked to the breaking of centro-symmetry. The onset of the K-edge 
+spectrum involves low-energy empty states on the cation that are typically of 3d orbital character. 
+The s→d transition is formally forbidden for an ideal, centro-symmetric octahedral motif. In lower 
+coordination motifs, e.g., tetrahedral, as well as distorted atomic motifs, hybridization with cation 
+4p orbitals leads to dipole-allowed transitions.[37] The pre-edge spectral trends associated with CN 
+and NNRS are a consequence of the breaking of centro-symmetry. This is consistent with the 
+original observations of Farges et al., and subsequent work.[31,37] 
+Drawing on the multiple scattering representation of the absorption process (Figure 1a), the 
+number of scatterers is an important factor that affects the absorption intensity, while path lengths 
+and interference effects determine the energy position where the absorption intensity changes. The 
+changes in main peak position, intensity, and shape are a key part of the spectral trend for both CN 
+and OCN (Figure 5c, z2 and z3). The overall increase of intensity with increasing coordination 
+number tracks the number of scatterers. For CN, the sharpening of the main peak shape with more 
+weight at lower energy as well as the shift of the post peak features to lower energy are further 
+examples of Natoli’s rule. As coordination increases, so do the metal-oxygen bond lengths. 
+Correspondingly, the path lengths get longer.  
+The OCN trend (Figure 5c, z3) modulates the intensity around the main peak and to a lesser 
+extent the post-peak region. The OCN descriptor correlates with the number of atoms in the second 
+
+18 
+ 
+coordination shell, and it covers a broader range of atoms that can contribute to three- and four-
+body scattering paths (Figure 1a). With an increase in the number of such paths induced by the 
+increase of such atoms, interference effects are stronger. But, as compared to the CN trend, the 
+energy location differs because of interatomic distances. OCN essentially counts the atoms in the 
+second coordination shell while CN counts the first coordination shell (Figure 4a), and hence is 
+related to larger interatomic distance. It results in a longer scattering path length (Figure 1a, SS2 
+versus SS1), longer wavelength, and consequently a lower energy photoelectron. As a result, the 
+OCN induces spectral variation only at lower energy while CN also induces variations at higher 
+energies (Figure 5c z3 vs z2). While OCN and second shell coordination number count the same 
+set of atoms, it is worth noting that the definition of cutoff radius for the second coordination shell 
+is ambiguous. In contrast, OCN is well-defined in established algorithms.[65] As a result, better 
+latent space-structure descriptor correlation can be observed for OCN (Figure 5c z3 versus Figure 
+5b z3) 
+The NNRS descriptor measures the spread in the cation-oxygen bond lengths in the first 
+coordination shell (Figure 4a). The scattering path lengths are spread out with larger NNRS. 
+Correspondingly, there is less reinforcement of constructive and destructive interference among 
+the paths as a function of photoelectron energy. This, in turn, reduces the definition of the peaks 
+and valleys in the spectrum. Specifically, as the NNRS increases, the peak-to-valley height 
+decreases. This is expressed, for example, in the local curvature in the post-peak spectral region, 
+quantified in the Cpit spectral feature (Figure 4b). The curvature is higher (deeper, better-defined 
+valley) when NNRS is smaller (Figure 5c z4). In the database and this analysis, a larger NNRS 
+value represents crystal structures with low symmetry and a spread of local bond lengths. It is 
+worth noting that Cpit is a specific spectral descriptor that can be clearly discerned for most 
+transition metals. Nonetheless, for specific metals, e.g., Mn in Figure S5b, the correlation between 
+Cpit and z4 is low with an SRCC of 0.05. As suggested by the spectral trends in Figure S8, the 
+curvature at the main peak correlates with z4 better where the SRCC increases to 0.33.  
+The variation in the local angles between the cation-oxygen bonds is another measure of spread 
+in low symmetry local motifs. It flows through to the path lengths for three-body scattering paths 
+(Figure 1a). In our comparison of different specific structure descriptors, we found that MOOD, 
+the distance minimum between two oxygens in the first coordination shell, was the most effective 
+metric for the RankAAE models (Figure 4a) and can be related to the 3-body scattering path 
+(Figure 1a, MS) in an X-ray absorption process. Based on a few examples in Figure S11, the 
+bending of the two axial oxygens towards the equatorial plane is one of the most intuitive structure 
+characteristics that the underlying latent variable z5 tracks. The distortion associated with the 
+bending is expected to change the 3-body scattering path length. The triangular path of MS is 
+relatively long and leads to oscillation at relatively low energy. In the case of vanadium oxides, it 
+is located around the main peak. With increasing oxygen-oxygen distance, the corresponding 
+wavelength increases and pushes the main peak position to lower energy.  
+V. CONCLUSION 
+ The approximately fifty-seven thousand simulated XANES K-edge data considered in this 
+study span eight 3d transition metal oxide families and exemplify complex data sets that domain 
+experts are highly motivated to study. Our novel rank constraint introduces physical 
+interpretability into the latent space representation of data produced by an AAE. Specifically, latent 
+variables are guided to track specific structure descriptors. From there, the reconstruction from 
+
+19 
+ 
+different regions in the latent space produces spectral variations reflecting the underlying changes 
+in structure descriptors. The correlations between structure descriptors and spectral trends are 
+confirmed to be both qualitatively and quantitatively observed in the ground truth data. 
+ For the specific example of XANES analysis, we have shown that our method significantly 
+extends the scope of spectral features a domain expert can analyze systematically, from local 
+metrics such as edge position to extended fingerprints that can encompass the full energy range of 
+XANES spectra. Leveraging the prior knowledge of domain experts, these spectral trends can be 
+used to gain further insight into features in materials structures and their relationship to spectral 
+trends. Also, as illustrated by the set of new structure descriptors explored in this work, it is 
+straightforward for a domain expert to explore new trends and new structure descriptors using the 
+RankAAE method driven by their intuition.  
+From a machine-learning perspective, we also highlight the challenges associated with the 
+extreme diversity of our datasets. First, the data is completely unbiased towards the specific 
+methods used in this study. It consists of materials data compiled independently by contributors to 
+the Materials Project over many years, driven by goals unrelated to the scope of this work. Second, 
+the target signal, namely a specific transition metal K-edge XANES spectrum, is sampled from a 
+set of diverse crystal structures with widely varying, albeit physically realistic, local structure 
+motifs and associated atomic species. Consequently, the data encompasses a broad variety of 
+physical complexities that affect the spectrum. These include different local chemical states of the 
+transition metal cation, scattering amplitudes that differ among atomic species, and the distinct set 
+of scattering paths (number of paths contributing, number of atoms in each path, and overall length 
+of each path) associated with each cation structure motif. This means that our methods have 
+numerically parsed a tremendous amount of diversity to identify the key trends we present. Thus, 
+the successes that we demonstrate in this work are quite encouraging and highlight the potential 
+of this, and related methods, for unearthing new spectral trends for impact on physical science and 
+engineering. 
+In summary, we demonstrate the capability of RankAAE using the XANES structure-spectrum 
+relationship as an example. However, the RankAAE is a general framework for enhancing the 
+interpretability of an AAE. In principle, it is also applicable to other kinds of spectroscopic data, 
+and in general, to any dataset in which one is attempting to discover physical correlations between 
+target signals and associated driving characteristics. 
+VI. METHODS  
+A. Structure of RankAAE and Training 
+AAE,[53] VAE[58] and WAE[9] regularize the latent space to a target distribution. The VAE 
+constrains its latent space by a KL divergence penalty to impose a prior distribution. The WAE 
+performs similar regularization by a Wasserstein distance.[66] The AAE does not require a specified 
+functional form of the penalty function to the prior distribution. It instead matches the aggregated 
+posterior of the latent vector with the prior distribution by a discriminator via an adversarial 
+training procedure. The AAE is reported to generate a better data manifold.[53,56,67,68] In this work, 
+a Gaussian distribution is used. We have adopted the gradient reversal layer from the domain 
+adversarial neural network to implement the adversarial training mechanism.[69] 
+The encoder, decoder, and discriminator (Figure 3) are all composed of 5 fully connected layers, 
+
+20 
+ 
+with a hidden layer size of 64. Both Dropout and Batch Normalization are used. A Parametric 
+Rectified Linear Unit (PReLU)[70] function is added after every fully connected layer with two 
+exceptions: (1) there is no activation function after the last layer in the encoder; (2) the last 
+activation function in the decoder is Softplus. 
+We incorporate the mutual information regularization term from the Dual Adversarial 
+AutoEncoder[71] to maximize the information shared between the latent space and the spectrum 
+data. We have also used a smooth loss in the early stage of the training to speed up the convergence 
+of network parameters. 
+With the performance evaluated on the validation set, the hyperparameters (learning rate, batch 
+size, noise-based data augmentation, gradient reversal amplitude, hidden layer size, the number of 
+hidden layers, and other aspects of the network structure) are optimized for vanadium oxides. The 
+performance is balanced between the appearance of the spectral trends and the correlation of the 
+latent variables with structure descriptors. 
+B. Rank Constraint 
+A direct regularization of latent space using a structure descriptor value will inevitably distort 
+the distribution. To avoid such distortion, we developed a novel regularization term based on the 
+Kendall rank correlation coefficient (KRCC).[72] It enforces a monotonic dependence while being 
+a soft enough constraint to allow variation in the final value. KRCC is built upon the concept of 
+concordant and discordant pairs. In the application to RankAAE, a latent variable for spectrum i 
+can be denoted as zi and the corresponding structure descriptor can be denoted as pi. Any pair of 
+(zi, pi) and (zj, pj) are said to be concordant if both zi > zj and pi > pj or both zi < zj and pi < pj; 
+otherwise, they are said to be discordant. KRCC describes the monotonic dependence by the ratio 
+of the number of concordant/discordant pairs. The counting is simplified using a sgn() function, 
+i.e., sgn (zi - zj) and sgn(pi - pj). However, the gradient of the sgn() function is 0 except at 0, which 
+is detrimental to the purpose of neural network parameter optimization. We have modified KRCC 
+to a loss function with a proper gradient everywhere by removing sgn() function around (zi - zj). 
+The loss function 𝐿𝑐 𝑧  for each constrained dimension of the latent space is defined by: 
+ 
+𝑓(𝑧𝑖, 𝑧𝑗) = (𝑧𝑖 − 𝑧𝑗)sgn(𝑝𝑖 − 𝑝𝑗) 
+ 
+ 
+ 
+N on = ∑ ∑ Θ (𝑓(𝑧𝑖, 𝑧𝑗))
+𝑛
+𝑗=1
+𝑛
+𝑖=1
+ 
+ 
+ 
+ 
+Ndis = ∑ ∑ Θ (−𝑓(𝑧𝑖, 𝑧𝑗))
+𝑛
+𝑗=1
+𝑛
+𝑖=1
+ 
+ 
+ 
+ 
+𝑓𝑐𝑜𝑛 𝑧 =
+ 
+m x[N on,  Ndis] ∑ ∑ Θ (𝑓(𝑧𝑖, 𝑧𝑗)) 𝑓(𝑧𝑖, 𝑧𝑗)
+𝑛
+𝑗=1
+𝑛
+𝑖=1
+ 
+ 
+ 
+𝑓𝑑𝑖𝑠 𝑧 =
+ 
+Ndis
+∑ ∑ Θ (−𝑓(𝑧𝑖, 𝑧𝑗)) 𝑓(𝑧𝑖, 𝑧𝑗)
+𝑛
+𝑗=1
+𝑛
+𝑖=1
+ 
+ 
+ 
+
+21 
+ 
+ 
+𝐿𝑐 𝑧 = −
+ 
+n n −   Ndis(𝑓𝑐𝑜𝑛 𝑧 + 𝑓𝑑𝑖𝑠 𝑧 ) 
+ 
+(1) 
+where z is a latent variable, p is a structure descriptor, n is the number of samples in a mini-
+batch in network training, i and j are sample indices in a minibatch, sgn() is a function to extract 
+the sign of a real number, Θ() is a Heaviside step function being 1 for positive values and 0 for 
+other values. Abbreviations “con” and “dis” designate concordant and discordant. Ncon and Ndis 
+represent the number of concordant and discordant pairs in the minibatch for the constrained 
+dimension of the latent space. The minimum value of Ndis,m is clamped to 1 to avoid the singularity. 
+The prefactors for the concordant and discordant pair contributions to the loss function are scaled 
+to be comparable early in the training and then to have a diminishing impact as more data points 
+become concordant. Overall, we find that this approach minimizes the effect of rank constraint on 
+the distribution of the latent space variable. 
+C. Data Acquisition and Preprocessing 
+The crystal structures are pulled from the Material Project.[54] Each crystal structure is a local 
+minimum in energy with lattice parameters and atomic coordinates.[73] All the crystal structures 
+containing each target transition metal atom and oxygen are included. The XANES K-Edge spectra 
+are simulated using the FEFF9 program[61] for the resulting database of about forty thousand 
+unique atomic sites identified in the oxides for the eight transition metals considered here with the 
+condition that the first neighbor shell of the transition metal site only contains oxygen. The 
+breakdown of the dataset according to transition metal is given in Table S1. 
+ In this work, XANES is defined as the spectrum with energy ranging from the start of the edge 
+of the spectrum to 50 eV above the edge. To create feature vectors that are directly comparable for 
+each metal considered, the spectra are interpolated to a fixed energy grid, equally spaced with 256 
+points. For machine learning, each data set is partitioned with a random selection of 70%, 15%, 
+and 15% of the spectra to form training, validation, and test sets, respectively. The size of the sets 
+is given in Table S1 for each transition metal. 
+D. Structure Descriptor Calculation 
+Eight structure descriptors have been explored in this study: 
+1) OS: oxidation state of a cation represented by an integer. OS is determined using the 
+maximum a posteriori (MAP) estimation method in pymatgen.[74–76] 
+2) CN: coordination number in the first coordination shell around a specified transition metal 
+cation. CN is determined using the CrystalNN algorithm in the pymatgen package.[65,75]  
+3) CN-2: coordination number in the second coordination shell for each target transition metal 
+cation. CN-2 is estimated by the number of atoms that falls in the layer between radius r1 
+and r2, chosen for each transition metal based on the radial distribution of the atoms. 
+4) OCN: oxygen coordination number. OCN is computed as an average of the coordination 
+numbers for the nearest neighbor oxygen atoms to the target transition metal cation. The 
+coordination number of each oxygen atom is determined using the CrystalNN algorithm in 
+the pymatgen package.[65,75]  
+5) NNRS: the standard deviation of the bond lengths from the target transition metal cation 
+to the nearest neighbor oxygen atoms. 
+
+22 
+ 
+6) NNAS: the standard deviation of the bond angles with the cation as the vertex and two 
+nearest neighbors as endpoints. 
+7) MOOD: the minimum oxygen-oxygen distance on the edges of the nearest neighbor 
+polyhedron encompassing the target transition metal cation. The polyhedron is determined 
+by the CrystalNN algorithm in the pymatgen package.[65,75] 
+8) PSGO: the point group symmetry order (PGSO) represented by an integer, computed as 
+the total number of symmetry operations for the point group for the full crystal space group 
+determined by the pymatgen package.[65,75] 
+E. Spectrum Descriptor Calculation 
+The spectrum descriptors shown in Figure 4b are defined as: 
+1) Eedge: the position of the absorption edge. 
+2) Ipre: pre-peak intensity, the maximum intensity among the peaks occurring at an energy 
+lower than the edge. 
+3) Emain: main peak position, the energy value at the first peak after the edge. 
+4) Imain: main peak intensity. For numerical stability purposes, we use the average over a 1 eV 
+window centered at Emain. 
+5) Cpit: post edge curvature. The position of the pit is found by a local minimum of the 
+spectrum in the portion at least 20 eV above the edge. The curvature is computed as the 
+second order derivative. To avoid numerical instabilities, the curvature is averaged over a 
+10 eV window centered at the pit.  
+6) Epost: second peak position. A second peak is identified by a local maximum of the spectrum. 
+To avoid ambiguities in peak counting, we considered Eedge + 15 eV for transition metals 
+Ti through Mn and Eedge + 20 eV for Fe through Cu as a proxy for Epost. The numerical test 
+in Figure S5b confirms the effectiveness of this approach.  
+7) Ipost: absorption intensity at Epost.  
+F. Spectral Variation Plot Generation 
+Specific latent space samples are fed to the decoder to generate reconstructed spectral variation 
+sequences. For each latent variable, 50 points on an equally spaced grid are sampled. From the 
+range of the latent space values encoded from the test set spectra, the latent space grid is chosen to 
+extend from the 5th  to the 95th  percentile values. For Figure S2a, a spectrum is reconstructed for 
+each of the 50 values of the specified latent space variable, setting the others equal to zero. For 
+Figure S2b and all of the other reconstructed spectral trends, an averaging procedure is applied to 
+sample the other latent space variables. Specifically, the average of reconstructed spectra for each 
+value of the target latent space variable is carried out over 10000 samples drawn from a 
+multivariant Gaussian distribution to fill the rest latent variables. There are 50 spectra in each 
+spectral trend presented. 
+G. Model Selection 
+For each dataset, we repeat the training 100 times with different random seeds for the neural 
+network parameters, and all the discussions are based on the best one. The performance for each 
+model is assessed by: 
+
+23 
+ 
+𝑆 = − m x
+𝑖,𝑗 |𝜌𝑖𝑗| + ∑|𝜌𝑖
+′|
+𝑖
+ 
+(2) 
+where 𝜌𝑖𝑗  is the Spearman rank correlation coefficient (SRCC) between a guided latent 
+variable 𝑧𝑖 and an unconstrained latent variable 𝑧𝑗, 𝜌𝑖
+′ is the correlation score (F1 score for OS and 
+CN, SRCC for other structure descriptors) between a constrained latent variable 𝑧𝑖 and the target 
+structure descriptor. To drive the performance metrics to have equal contributions, the z-scores are 
+computed across the 100 models by subtracting the average and dividing the standard deviation 
+for each 𝜌𝑖𝑗 and 𝜌𝑖
+′. Then the z-scores[77] are used to evaluate S in Eq. (2). A good model maximizes 
+the latent-structure correlation and minimizes the intercorrelation between the unconstrained latent 
+variables and the other dimensions. Therefore, a larger S indicates better performance. For this 
+study, it is sufficient to consider the two terms with equal weight. However, for other studies, the 
+assessment of model performance can be further refined by scaling the two terms differently. Table 
+S2 details the statistics of different loss terms (prior to z-score normalization) that enter Eq. (2), as 
+well as the standard reconstruction error attributed to the autoencoder. For the current study, the 
+reconstruction error is small and it does not vary significantly. Hence, we did not include 
+reconstruction error in the model selection. The other terms vary within reasonable bounds. For 
+further insight, the correlation plots for the worst model out of 100 runs by this criterion for the 
+case of V oxide are shown in Figure S12.  Encouragingly, the correlations for that model are still 
+reasonably good. 
+ACKNOWLEDGMENT 
+This research is based upon work supported by the U.S. Department of Energy, Office of 
+Science, Office Basic Energy Sciences, under Award Number FWP PS-030. This research also 
+used the Theory and Computation facility of the Center for Functional Nanomaterials (CFN), 
+which is a U.S. Department of Energy Office of Science User Facility, at Brookhaven National 
+Laboratory under Contract No. DE-SC0012704. 
+SUPPORTING INFORMATION 
+Supporting Information is available. 
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+ 
+ 
+
+27 
+ 
+Table of Contents 
+A new approach to harness powerful machine learning methods based on autoencoders results 
+in compact, physically interpretable representations of complex spectral datasets. This paradigm 
+shift enables discovery of structure-spectrum relationships, applicable to a wide range of scientific 
+fields. A showcase study demonstrates new relationships to extract more structure information 
+from X-ray absorption spectra. 
+ 
+ 
+Authors: Zhu Liang, Matthew R. Carbone, Wei Chen, Fanchen Meng, Eli Stavitski, Deyu Lu,* 
+Mark S. Hybertsen,* and Xiaohui Qu* 
+Title: Decoding Structure-Spectrum Relationships with Physically Organized Latent Spaces 
+ 
+ 
+ 
+z
+Encoder
+Decoder
+Physical Latent Space
+Coordination
+Oxidation
+
+28 
+ 
+Supporting Information for “Decoding Structure-Spectrum 
+Relationships with Physically Organized Latent Spaces” 
+ 
+Zhu Liang,1 Matthew R. Carbone,2 Wei Chen,1 Fanchen Meng,1 Eli Stavitski,3 Deyu Lu,1,* Mark S. 
+Hybertsen,1,* and Xiaohui Qu1* 
+ 
+1Center for Functional Nanomaterials, Brookhaven National Laboratory, Upton, New York 11973, 
+USA  
+2Computational Science Initiative, Brookhaven National Laboratory, Upton, New York 11973, 
+USA  
+3National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, 
+USA 
+* [email protected], [email protected], [email protected] 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+ 
+
+29 
+ 
+ 
+Figure S1. Comparing the generative characteristics for (a) a regular autoencoder (AE) model 
+compared to (b) an adversarial autoencoder (AAE) model, each trained independently on the 
+vanadium oxide data set. The dimension of the latent space is set to 6, representative of other 
+models considered in this study. The decoded spectra are sampled in five cohorts of 10 randomly 
+sampled data points in the latent space with each cohort displayed in a separate row. The main text 
+highlights differences observed for AE versus AAE trained models in the literature. Here the 
+consequence shortcomings of the AE model are illustrated for the exemplary XANES data 
+specifically.  
+   
+           
+                      
+   
+           
+                      
+
+30 
+ 
+ 
+ 
+Figure S2. Spectral trends in vanadium oxides computed using the decoder in the RankAAE model 
+by two different procedures: (a) Hold other latent dimensions to zero. (b) Sample and average 
+other latent dimensions as described in Methods. Part (b) is duplicated from main text, Figure 5c. 
+ 
+   
+   
+            
+                       
+
+Z1~OS: 0.91
+Z2~CN: 0.96
+Z3~OCN:0.86
+Z4~NNRS:0.90
+Z5~MOOD:0.86
+Z6~Unguided:N/A
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+H
+H
+H
+L
+IH
+H
+HZ1~OS: 0.91
+Z2~CN: 0.96
+Z3~OCN: 0.86
+Z4~NNRS:0.90
+Z5~MOOD:0.86
+Z6~Unguided:N/A
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+H
+H
+H
+H
+H
+H31 
+ 
+ 
+ 
+Figure S3. Spectral trends from a series of independently trained RankAAE models constrained 
+with different sets of structure descriptors in each row as noted for each trend. Starting in the first 
+row (a) with descriptors {CS, CN, CN-2, NNAS, PGSO}, progressively one descriptor is replaced 
+to arrive at the final set {CS, CN, OCN, NNRS, MOOD} in the last row (d). The annotated label 
+for each spectral trend refers to the related latent variable, structure descriptor, and the F1 score or 
+the Spearman rank correlation coefficient (SRCC) between them. 
+ 
+ 
+
+(a)
+Z1~OS: 0.89
+Z2~CN: 0.95
+Z3~CN2: 0.83
+Z4~NNAS: 0.82
+Z5~PGSO: 0.68
+Z6~Unguided:N/A
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+IH
+H
+IH
+IH
+L
+H
+L
+H
+(b)
+Z1~OS: 0.92
+Z2~CN: 0.96
+Z3~OCN: 0.88
+Z4~NNAS: 0.80
+Z5~PGSO: 0.65
+Z6~Unguided:N/A
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+JH
+JH
+H
+H
+(c)
+Z1~OS: 0.91
+Z2~CN: 0.96
+Z3~OCN: 0.88
+Z4~NNRS: 0.88
+Z5~PGSO: 0.65
+Z6~Unguided:N/A
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+JH
+L
+IH
+IH
+JH
+JH
+(d)
+Z1~OS: 0.91
+Z2~CN: 0.96
+Z3~OCN: 0.86
+Z4~NNRS:0.90
+Z5~MOOD: 0.86
+Z6~Unguided:N/A
+Z1
+Z2
+Z3
+Z4
+Z5
+Z6
+H
+L
+H
+H
+TH
+L
+H
+H32 
+ 
+ 
+ 
+Figure S4. Series of independently trained RankAAE models showing the evolution of the models 
+as structure descriptors are added sequentially, starting with OS. In addition, to the unguided latent 
+space variable, we include another dimension guided to reproduce a pseudo-random number 
+sequence as a structure descriptor (annotated as NOISE).  In the top row, the dimension of the 
+latent space is N=3. A dimension is added successively for each additional structure descriptor 
+until N=7 in the final row.  
+ 
+ 
+ 
+ 
+           
+                      
+
+Z1~OS: 0.87
+Z2~NOISE:0.03
+Z3~Unguided: N/AZ1~OS:0.88
+Z2~CN:0.94
+Z3~NOISE:0.05
+Z4~Unguided:N/AZ1~OS:0.88
+Z2~CN:0.95
+Z3~OCN: 0.89
+Z4~NOISE:0.02
+Z5~Unguided: N/AZ1~OS:0.87
+Z2~CN: 0.95
+Z3~OCN: 0.88
+Z4~NNRS:0.87
+Z5~NOISE:0.05
+Z6~Unguided:N/AZ1~OS:0.88
+Z2~CN: 0.96
+Z3~OCN:0.88
+Z4~NNRS:0.88
+Z5~MOOD:0.86
+Z6~NOISE: -0.00
+Z7~Unguided:N/A33 
+ 
+ 
+ 
+Figure S5. F1 score (z1~OS and z2~CN) and SRCC (z3~OCN, z4~NNRS and z5~MOOD) assessing 
+the correlation of latent variable to (a) the target structure descriptors and (b) the primary spectral 
+descriptors for each of the RankAAE models trained independently for each transition metal oxide 
+data set.  The test portion of the data set is used for this assessment. The correlation between latent 
+variable z5 and spectral descriptor Emain is computed for each CN category and the average value 
+is shown in (b).  
+ 
+ 
+ 
+   
+   
+                 
+                 
+
+Z1
+z3
+z5
+72
+z434 
+ 
+ 
+ 
+Figure S6. Illustration of multiple contributions to the main peak position spectral descriptor Emain 
+(y-axis) for the test set portion of the vanadium oxide data set. Contribution from CN is shown by 
+color (CN4: blue, CN5: orange, CN6: green), and contribution from latent variable z5 is shown by 
+the x-axis. The top figure (a) shows the data in aggregate while the bottom role of figures (b-c) 
+separates out the data according to CN.  
+ 
+ 
+   
+   
+   
+   
+
+35 
+ 
+ 
+ 
+Figure S7. Full data set of K-edge XANES spectra for each of the transition metal oxides showing 
+the full range of spectral variation. Each row is colored by a specific structure descriptor annotated 
+in the plots: CS, CN, OCN, NNRS, MOOD. The bottom row is the full data set with no further 
+distinction. 
+
+36 
+ 
+ 
+ 
+Figure S8. Spectral trends for independently trained RankAAE models for each of the transition 
+metal oxide data sets using the final set of descriptors determined for the specific case of vanadium 
+oxides and presented in Figure 5c. The format of the plot follows Figure 5c in the main text. 
+
+37 
+ 
+ 
+ 
+
+38 
+ 
+ 
+ 
+Figure S9. Correlation between the RankAAE latent space (x-axis), structure descriptor (y-axis), 
+and spectral descriptor (color) for the each of the transition metal oxides data sets. The format of 
+the plots follows Figure 6 in the main text.  
+ 
+
+39 
+ 
+ 
+ 
+Figure S10. Alternative representation of the spectral fingerprint associated with each spectral 
+trend derived from the sequence of RankAAE models constrained to different descriptors and 
+shown in Figure S1. Shown is the difference between extremal spectra for each trend (95th and 5th 
+percentile of the latent values). The scale for each box is identical, both amplitude (y-axis) and 
+energy scale (x-axis), so the visual comparison of amplitude is meaningful. To assess overlaps, a 
+matrix of cosine similarities is computed for the fingerprints in each row. The value of the largest 
+overlap is annotated by the “max_cos_sim” score and the specific style with the largest overlap is 
+noted.   
+ 
+ 
+           
+                                
+
+max_cos_sim:0.55
+max_cos_sim:0.17
+max_cos_sim:0.55
+max_cos_sim:0.58
+max_cos_sim:0.58
+max_cos_sim:0.29
+with style3
+with style5
+with style1
+with style5
+withstyle4
+with style3
+Z1~OS: 0.91
+Z2~CN: 0.96
+Z3~OCN:0.86
+Z4~NNRS:0.90
+Z5~MOOD:0.86
+Z6~Unguided: N/Amax_cos_sim:0.64
+max_cos_sim:0.32
+max_cos_sim:0.60
+max_cos_sim:0.48
+max_cos_sim:0.64
+max_cos_sim:0.61
+withstyle5
+with style3
+withstyle6
+withstyle5
+with style1
+with style1
+Z1~OS: 0.91
+Z2~CN: 0.96
+Z3~OCN:0.88
+Z4~NNRS:0.88
+Z5~PGSO:0.65
+Z6~Unguided: N/Amax_cos_sim:0.66
+max_cos_sim:0.26
+max_cos_sim:0.55
+max_cos_sim:0.22
+max_cos_sim:0.66
+max_cos_sim:0.51
+withstyle5
+with style3
+with style1
+with style5
+with style1
+with style3
+Z1~OS: 0.92
+Z2~CN:0.96
+Z3~OCN:0.88
+Z4~NNAS:0.80
+Z5~PGSO:0.65
+Z6~Unguided: N/Amax_cos_sim:0.82
+maxlcos_sim:0.68
+max_cos_sim:0.68
+max_cos_sim:0.50
+max_cos_sim:0.82
+max_cos_sim:0.53
+withstyle5
+with style3
+with style2
+with style5
+with style1
+with style1
+Z1~OS: 0.89
+Z2~CN:0.95
+Z3~CN2:0.83
+Z4~NNAS:0.82
+Z5~PGSO:0.68
+Z6~Unguided: N/A40 
+ 
+ 
+ 
+Figure S11. Example atomic motifs associated with different ranges of latent variable z5 value. 
+The material project ID, value of latent variable z5, angle between the cation bonds to the two axial 
+oxygen  (θ), and structure descriptor MOOD are annotated below each motif. 
+ 
+z5 = -1.27
+θ = 152.0 
+MOOD = 2.56 Å
+mp-766107
+z5 = -1.48
+θ = 153.2 
+MOOD = 2.47 Å
+mp-1094019
+z5 = -1.75
+θ = 157.5 
+MOOD = 2.52 Å
+mp-704734
+(a) Low z5
+(c) High z5
+z5 = 0.94
+θ = 179.5 
+MOOD = 2.77 Å 
+mp-1076338
+z5 = 1.29
+θ = 179.6 
+MOOD = 2.75 Å
+mp-1075987
+z5 = 1.06
+θ = 180.0 
+MOOD = 2.79 Å 
+mp-19053
+O
+V
+(b) Middle z5
+z5 = -0.21
+θ = 165.6 
+MOOD = 2.56 Å
+mp-766372
+z5 = -0.31
+θ = 170.2 
+MOOD = 2.63 Å
+mp-760018
+z5 = -0.03
+θ = 176.0 
+MOOD = 2.69 Å
+mp-759891
+
+41 
+ 
+ 
+ 
+Figure S12. For each choice of descriptor set, the RankAAE model is trained 100 times. The 
+performance score S defined by Methods equation (2) on the test portion of the data set ranges 
+from -12.91 to 7.02.  The correlation plots are shown for the worst model with S=-12.91 for 
+vanadium oxide.  
+ 
+ 
+ 
+ 
+           
+          
+            
+    
+
+(a) Os
+Eedge
+IH
+(b) CN
+'pre
+H
+(c) OCN
+(d) NNRS
+(e) MOOD
+Imain
+Cpit
+Emain
+H
+H(p)
+Z1~OS: 0.92
+Z2~CN:0.94
+Z3~OCN: 0.82
+Z4~NNRS:0.86
+Z5~MOOD:0.80
+Z6~Unguided:N/A
+Z1
+Z2
+Z3
+Z4
+Zs
+Z6
+H
+LI
+IH
+IH
+H
+H
+L
+H42 
+ 
+Table S1. Size of datasets for all metals.  
+ 
+Total 
+Training 
+Test 
+Validation 
+Ti 
+6488 
+4541 
+973 
+974 
+V 
+9312 
+6518 
+1396 
+1398 
+Cr 
+3596 
+2517 
+539 
+540 
+Mn 
+11755 
+8228 
+1763 
+1764 
+Fe 
+7446 
+5212 
+1116 
+1118 
+Co 
+10146 
+7102 
+1521 
+1523 
+Ni 
+3666 
+2566 
+549 
+551 
+Cu 
+4564 
+3194 
+684 
+686 
+ 
+ 
+ 
+
+43 
+ 
+Table S2. Statistical characterization of performance metrics across 100 independently trained 
+models for the case of vanadium oxide. The model selection process utilizes values further 
+normalized by z-scores (Methods). Metrics are computed on the test portion of the data set. 
+ 
+Average 
+Standard 
+Deviation 
+Minimum 
+Maximum 
+𝜌𝑖𝑗 
+0.277 
+0.092 
+0.049 
+0.556 
+Reconstruction Error* 
+0.037 
+0.001 
+0.034 
+0.041 
+ 
+𝜌1
+′: z1~OS 
+0.890 
+0.008 
+0.876 
+0.907 
+ 
+𝜌2
+′: z2~CN 
+0.946 
+0.010 
+0.911 
+0.964 
+ 
+𝜌3
+′: z3~OCN 
+0.860 
+0.011 
+0.821 
+0.881 
+ 
+𝜌4
+′: z4~NNRS 
+0.884 
+0.016 
+0.842 
+0.910 
+ 
+𝜌5
+′: z5~MOOD 
+0.840 
+0.012 
+0.805 
+0.863 
+* The reconstruction error is computed as the mean absolute deviation (MAD) between the original 
+spectrum and reconstructed spectrum 
+ 
+ 
+